The lattice of abstract elementary classes of modulesThanks: Research of Marcos Mazari-Armida was partially supported by NSF grant DMS-2348881 and Simons Foundation grant MPS-TSM-00007597. Research of Gianluca Paolini was supported by project PRIN 2022 “Models, Sets and Classifications”, prot. 2022TECZJA, and by INdAM Project 2024 (Consolidator grant) “Groups, Crystals and Classifications”.

Tapani Hyttinen Address: Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68 (Pietari Kalmin katu 5), FI-00014 University of Helsinki, Finland. Email address: tapani.hyttinen@helsinki.fi , Marcos Mazari-Armida Address: Department of Mathematics, Baylor University, Sid Richardson Building, 1410 S. 4th Street, Waco, TX 76706, USA Email address: marcos_mazari@baylor.edu URL: https://sites.baylor.edu/marcos_mazari/ and Gianluca Paolini Address: Department of Mathematics “Giuseppe Peano”, University of Turin, Palazzo Campana, Via Carlo Alberto 10, 10123 Torino, Italy. Email address: gianluca.paolini@unito.it URL: https://sites.google.com/view/gianlucapaolini/

Abstract.

Let $R$ be a ring. We organize the abstract elementary classes whose underlying class is the class of all $R$ -modules and whose strong submodel relation lies between the submodule and direct summand relations into a lattice $\mathscr{L}_{R}$ , ordered by reverse inclusion. We establish the basic lattice-theoretic properties of $\mathscr{L}_{R}$ and investigate its two natural sublattices, below and above purity. Below purity, we isolate relations defined by first-order $\mathrm{pp}$ -formulas for which amalgamation, tameness, and stability hold. Above purity, we introduce relations defined by infinitary $\mathrm{pp}$ -formulas and prove a broad stability result. Specializing to abelian groups, we show that the lattice $\mathscr{L}_{\mathbb{Z}}$ has the following properties: it has a strong submodel relation that is not positive syntactic, it contains an uncountable antichain and a strictly increasing proper-class-sized chain, and it has a broad region above purity where amalgamation fails.

Key words and phrases:

Abstract elementary classes; Modules; Stability; Amalgamation.

2020 Mathematics Subject Classification

Primary: 03C48, 03C60; Secondary: 03C45, 16D90, 13L05, 20K27.

1. Introduction

Abstract elementary classes (AECs for short) constitute the prominent model-theoretic framework used to study non-elementary classes of structures. These were introduced in the seventies by Shelah [26]. For many years the focus was on developing the abstract theory [1, 4, 30], but recently there has been significant activity on finding interactions and applications of AECs to module theory [3, 5, 18, 19, 21, 24]. This paper introduces a new kind of interaction between AECs and module theory which we expect will be very fruitful.

An abstract elementary class is a pair $\mathcal{K}=(\mathbf{K},\preccurlyeq)$ , where $\mathbf{K}$ is a class of structures and $\preccurlyeq$ is a partial order on $\mathbf{K}$ , subject to a few natural axioms (see Definition 2.14). Two of the key axioms are that $\mathcal{K}$ is closed under directed colimits and that every element of $\mathbf{K}$ can be decomposed into small elements of $\mathbf{K}$ . In this paper, $\mathbf{K}$ will always be a class of modules over a fixed ring $R$ ; in most cases it will in fact be the class of all $R$ -modules.

Most of the interactions of AECs and module theory have taken place by fixing a class of modules and studying the AEC one obtains by letting the partial order on the class be the pure submodule relation¹¹ 1 $A$ is a pure submodule of $B$ if every finite system of $R$ -linear equations with the right-hand side in $A$ that is solvable in $B$ is already solvable in $A$ .. The choice of purity as the de facto partial order to study classes of modules is motivated both by the algebraic importance of purity [9, 10] and the first-order model-theoretic result of $\mathrm{pp}$ -quantifier elimination [22]. Nevertheless there is no deep reason why one should restrict to purity and recently interesting results for other partial orders have been obtained [7, 19, 20, 21].

In this paper, we fix the class of all modules and instead of studying the AEC one obtains by fixing the partial order to be purity (as it was done in [17]) or another partial order, we study all the possible partial orders that give an AEC on the class of all modules.

This new approach has two key benefits. The first is that it allows us to obtain results for a family of AECs instead of a single AEC. This lets us recover many results from the literature (see Remarks 4.18 and 4.42). The second, which we find to be the most significant, is that it allows us to uncover a new mathematical object $\mathscr{L}_{R}$ , which is the main object of study of this paper. $\mathscr{L}_{R}$ is the lattice of binary relations $\preccurlyeq$ on the class of $R$ -modules such that the class of $R$ -modules equipped with $\preccurlyeq$ is an abstract elementary class and $\preccurlyeq$ lies between the submodule and direct summand relations, ordered by reverse inclusion (see Definition 3.1 and Corollary 3.5). Since the underlying class is fixed and only the strong submodel relation varies, $\mathscr{L}_{R}$ measures how different algebraically natural notions of submodule change the model-theoretic behavior of the class of modules.

The results on the lattice are of two sorts. The first consists of isolating properties that allow us to show that a broad region of the lattice has or fails to have important model-theoretic properties. The second consists of proving lattice-theoretic results. These results depend heavily on the structure of the ring $R$ ; for instance, the lattice is trivial if $R$ is semisimple. Thus, in this paper we focus mostly on the case when $R=\mathbb{Z}$ .

We divide the study of the lattice into two natural sublattices: the sublattice below purity $\mathscr{L}^{1}_{R}$ and the sublattice above purity $\mathscr{L}^{2}_{R}$ . The techniques we employ to study $\mathscr{L}^{1}_{R}$ mimic those used for purity, while the techniques we employ for $\mathscr{L}^{2}_{R}$ are new. Regarding $\mathscr{L}^{2}_{R}$ , it is worth noticing that even the existence of elements in $\mathscr{L}^{2}_{R}$ strictly extending purity was not known before this work. We construct a plethora of elements in $\mathscr{L}^{2}_{R}$ by using infinitary $\mathrm{pp}$ -formulas (see Definition 2.1).

The relations above purity, in $\mathscr{L}^{2}_{R}$ , measure the extent to which solvability of infinite systems of linear equations in the ambient module is reflected in the submodule. The relation $\leqslant^{\mathrm{weak}}_{\mathrm{pp,\infty}}$ requires the two modules to agree on every weak infinitary $\mathrm{pp}$ -formula; equivalently, whenever a possibly infinite family of first-order $\mathrm{pp}$ -formulas over a finite tuple from the smaller module is realized by a finite tuple in the larger module, it is already realized in the smaller one. The relations $\leqslant^{(\kappa,\alpha)}_{\mathrm{pp}}$ and $\leqslant^{(\kappa,\infty)}_{\mathrm{pp}}$ refine the same idea by controlling the size and the syntactic complexity of the admissible infinitary conjunctions (see Definition 4.22). This connects with the study of $\mu$ - $\mathrm{AEC}$ s of modules initiated in [6]. The relations of $\mu$ -purity considered there, however, fail to give rise to $\mathrm{AEC}$ s, so the relations $\leqslant^{\mathrm{weak}}_{\mathrm{pp,\infty}}$ , $\leqslant^{(\kappa,\alpha)}_{\mathrm{pp}}$ and $\leqslant^{(\kappa,\infty)}_{\mathrm{pp}}$ introduced here take on particular interest as they are genuinely new.

The main model-theoretic properties studied in this paper are stability and the amalgamation property. Stability is one of the key dividing lines in model theory (see Definition 2.18); it was introduced for elementary classes in [25] and for AECs in [27]. We show that stability is pervasive in both $\mathscr{L}^{1}_{R}$ and $\mathscr{L}^{2}_{R}$ . In passing, we recall that it was an open problem whether there exists an unstable AEC of modules with respect to purity; this was answered in the positive in [21] (see also [13]).

Theorem 1.1.

(1)

(Lemma 4.17) Assume that $\preccurlyeq\in\mathscr{L}^{1}_{R}$ is pleasant (see Definition 4.6). If $\lambda^{\operatorname{card}(R)+\aleph_{0}}=\lambda$ , then $(R\text{-}\mathrm{Mod},\preccurlyeq)$ is $\lambda$ -stable.
(2)

(Theorem 4.38) Assume that $\preccurlyeq\in\mathscr{L}^{2}_{R}$ is E-nice (see Definition 4.32). Then $(R\text{-}\mathrm{Mod},\preccurlyeq)$ is stable.

The proofs of the above two results are very different. The first result is obtained by first syntactically characterizing (Galois or orbital) types (see Theorem 4.12) and then counting the number of types using similar tools to the classical tools from first-order model theory. The second result uses in a key way the pushout construction and a local character-like result (see Claim 1 of Theorem 4.38).

Amalgamation is a classical assumption on AECs (see Definition 2.16). This is the case because amalgamation is seen as a weak consequence of the compactness theorem, and because it is expected to follow from categoricity in a tail of cardinals [11, Conjecture 2.3]. We show that amalgamation holds often in $\mathscr{L}^{1}_{R}$ , but often fails in $\mathscr{L}^{2}_{R}$ when $R=\mathbb{Z}$ .

Theorem 1.2.

(1)

(Lemma 4.9) Assume $\preccurlyeq\in\mathscr{L}^{1}_{R}$ is pleasant. Then $(R\text{-}\mathrm{Mod},\preccurlyeq)$ has the amalgamation property.
(2)

(Theorem 5.17) Assume $\preccurlyeq\in\mathscr{L}^{2}_{\mathbb{Z}}$ is positive syntactic (see Definition 5.4) and lies above $\leqslant^{\mathrm{weak}}_{\mathrm{pp,\infty}}$ (see Definition 4.26) in $\mathscr{L}_{\mathbb{Z}}$ . Then $(\mathbb{Z}\text{-}\mathrm{Mod},\preccurlyeq)$ does not have the amalgamation property.

The proofs of the above two results rely heavily on the pushout construction. The second result once again suggests [19, 29] that amalgamation is not as pervasive as commonly assumed. Furthermore, the second result shows that the model-theoretic behavior of the class of modules for a relation above purity is distinct from that for purity.

We obtain a couple of basic lattice-theoretic properties for $\mathscr{L}_{R}$ for an arbitrary ring $R$ (see Section 3.1). As mentioned earlier, since $\mathscr{L}_{R}$ depends so heavily on the ring-theoretic properties of $R$ , in this paper we focus on the case when $R=\mathbb{Z}$ . The results for $\mathscr{L}_{\mathbb{Z}}$ use classical tools from abelian group theory such as the notion of isotype subgroup.

Theorem 1.3.

(1)

(Proposition 4.3) $|\mathscr{L}^{1}_{\mathbb{Z}}|\leqslant 2^{2^{\aleph_{0}}}$ .
(2)

(Lemma 5.1) $\mathscr{L}^{1}_{\mathbb{Z}}$ has an uncountable antichain.
(3)

(Theorem 5.13) There is a strictly increasing proper-class-sized chain in $\mathscr{L}_{\mathbb{Z}}$ . In particular, the lattice $\mathscr{L}_{\mathbb{Z}}$ is a proper class.

We encourage the reader to look at Appendix A where we give a depiction of $\mathscr{L}_{\mathbb{Z}}$ .

Finally, a surprising result is that there is a relation $\preccurlyeq\in\mathscr{L}_{\mathbb{Z}}$ below purity such that $\preccurlyeq$ cannot be characterized by infinitary $\mathrm{pp}$ -formulas (see Theorem 5.5). This result is important as it shows that the study of $\mathscr{L}_{\mathbb{Z}}$ is not simply the study of binary relations generated by infinitary $\mathrm{pp}$ -formulas.

Taken together, these results reveal a global contrast: stability is widespread both below and above purity, while amalgamation fails on a broad region above purity, and already for abelian groups the resulting lattice is a proper class.

The paper is divided into five sections. Section 2 has some preliminaries and basic results, including an introduction to infinitary $\mathrm{pp}$ -formulas. Section 3 introduces the main object of study of this paper and presents some of its basic lattice-theoretic properties. Section 4 has important model-theoretic results for broad regions of the sublattices $\mathscr{L}^{1}_{R}$ and $\mathscr{L}^{2}_{R}$ . Section 5 contains several results on $\mathscr{L}_{\mathbb{Z}}$ .

2. Preliminaries and basic results

In this section, we present a quick introduction to both the model theory of modules and abstract elementary classes.

2.1. Model theory of modules

All rings considered in this paper are associative with identity, and all modules are left $R$ -modules unless stated otherwise. For a fixed ring $R$ , we denote the class of $R$ -modules by $R\text{-}\mathrm{Mod}$ . For $A,B\in R\text{-}\mathrm{Mod}$ , we write $A\leqslant B$ if $A$ is a submodule of $B$ , and $A\leqslant_{\oplus}B$ if $A$ is a direct summand of $B$ . We write $\operatorname{card}(R)$ for the cardinality of the ring $R$ .

Throughout this subsection, $R$ is a fixed ring. The language of $R$ -modules is $\tau_{R}=\{0,+,-\}\cup\{r\cdot\mid r\in R\},$ where for each $r\in R$ , the symbol $r\cdot$ is interpreted as scalar multiplication by $r$ .

Following [28, Definition 2.2], we introduce infinitary $\mathrm{pp}$ -formulas.

Definition 2.1.

Let $\lambda\leqslant\kappa$ be infinite cardinals. For every ordinal $\epsilon<\lambda$ , we define by induction on $\alpha$ a set $\Lambda_{\alpha,\epsilon}^{\kappa,\lambda}$ of formulas $\varphi(\bar{x})$ in $\mathfrak{L}_{\kappa,\lambda}(\tau_{R})$ with $|\bar{x}|=\epsilon$ as follows:

(1)

For $\alpha=0$ , we define $\Lambda_{\alpha,\epsilon}^{\kappa,\lambda}$ as the set of all equations of the form

$\sum_{\zeta<\epsilon}r_{\zeta}x_{\zeta}=0$

with $r_{\zeta}\in R$ and $r_{\zeta}\neq 0$ for only finitely many $\zeta<\epsilon$ .
(2)

For $\alpha$ a limit ordinal, we define $\Lambda_{\alpha,\epsilon}^{\kappa,\lambda}=\bigcup_{\beta<\alpha}\Lambda_{\beta,\epsilon}^{\kappa,\lambda}$ .
(3)

For $\alpha=\beta+1$ , we define $\Lambda_{\alpha,\epsilon}^{\kappa,\lambda}$ as the set of formulas of the form

$\varphi(\bar{x})=\exists\bar{y}\bigwedge\{\psi(\bar{x},\bar{y})\mid\psi(\bar{x},\bar{y})\in\Psi\}$

for some $|\bar{y}|=\zeta<\lambda$ ²² 2 We allow $\zeta=0$ . and some $\Psi\subseteq\Lambda_{\beta,\epsilon+\zeta}^{\kappa,\lambda}$ with $|\Psi|<\kappa$ .

Remark 2.2.

We are primarily interested in the case when $\lambda=\aleph_{0}$ .

Definition 2.3.

(1)

We define $\Lambda_{\alpha}^{\kappa,\lambda}=\bigcup_{\epsilon<\lambda}\Lambda_{\alpha,\epsilon}^{\kappa,\lambda}$ .
(2)

We define $\Lambda_{\alpha}^{\infty,\lambda}=\bigcup\{\Lambda_{\alpha}^{\kappa,\lambda}\,\mid\,\kappa\in\mathrm{Card}\}$ .
(3)

We define $\Lambda_{\infty}^{\kappa,\lambda}=\bigcup\{\Lambda_{\alpha}^{\kappa,\lambda}\,\mid\,\alpha\in\mathrm{Ord}\}$ .
(4)

We define $\Lambda_{\infty}^{\infty,\infty}=\bigcup\{\Lambda_{\alpha}^{\kappa,\lambda}\,\mid\,\alpha\in\mathrm{Ord},\kappa,\lambda\in\mathrm{Card}\}$ .

We say that $\varphi(\bar{x}),\psi(\bar{x})\in\mathfrak{L}_{\infty,\infty}(\tau_{R})$ are equivalent (or, more precisely, equivalent relative to the theory of $R$ -modules) if, for every $R$ -module $A$ and every $\bar{a}\in A^{|\bar{x}|}$ , we have $A\models\varphi(\bar{a})$ if and only if $A\models\psi(\bar{a})$ .

Definition 2.4.

$\varphi(\bar{x})$ is a first-order positive primitive formula (a $\mathrm{pp}$ -formula, for short) if it is a first-order formula in the language of $R$ -modules and is equivalent to a formula in $\Lambda_{\omega}^{\aleph_{0},\aleph_{0}}$ .

We say that $A$ is a pure submodule of $B$ , denoted by $A\leqslant_{\mathrm{pp}}B$ , if $A\leqslant B$ and, for every first-order $\mathrm{pp}$ -formula $\varphi(\bar{x})$ and every $\bar{a}\in A^{|\bar{x}|}$ , we have $A\models\varphi(\bar{a})$ if and only if $B\models\varphi(\bar{a})$ . Equivalently, every finite system of $R$ -linear equations with right-hand side in $A$ that is solvable in $B$ is already solvable in $A$ .

The following two results study the syntactic structure of infinitary $\mathrm{pp}$ -formulas.

Proposition 2.5.

Let $\varphi(\bar{x})\in\Lambda_{\alpha}^{\kappa,\aleph_{0}}$ , where $\bar{x}=(x_{0},\ldots,x_{n-1})$ , $\kappa$ is a cardinal, and $\alpha$ is an ordinal. Then $\varphi(\bar{x})$ is equivalent, relative to the theory of $R$ -modules, to a formula of the form

\exists\bar{v}\bigwedge_{i<\lambda}\delta_{i}(\bar{v},\bar{x})=0,

with $\lambda\leqslant\kappa$ ³³ 3 If $\kappa$ is regular, then $\lambda<\kappa$ ., $\bar{v}=(v_{i})_{i<\lambda}$ , and, for every $i<\lambda$ ,

\delta_{i}(\bar{v},\bar{x})\doteq\sum_{\ell<\lambda}r_{i,\ell}v_{\ell}+\sum_{j<n}q_{i,j}x_{j},

with $r_{i,\ell},q_{i,j}\in R$ and for each $i<\lambda$ , $r_{i,\ell}\neq 0$ for only finitely many $\ell<\lambda$ . In particular, $\varphi(\bar{x})$ is equivalent to a formula in $\Lambda^{\lambda^{+},\lambda^{+}}_{1}$ .

Proof.

The final assertion follows directly from the main statement. We prove by induction on $\alpha$ that the conclusion holds for every natural number $n$ and every $\varphi(\bar{x})\in\Lambda_{\alpha,n}^{\kappa,\aleph_{0}}$ .

If $\alpha=0$ or $\alpha$ is a limit ordinal, there is nothing to prove. Assume that $\alpha=\beta+1$ . Then

\varphi(\bar{x})=\exists\bar{y}\bigwedge\{\psi(\bar{x},\bar{y})\mid\psi(\bar{x},\bar{y})\in\Psi\}

for some $|\bar{y}|=m<\aleph_{0}$ and some $\Psi\subseteq\Lambda_{\beta,n+m}^{\kappa,\aleph_{0}}$ with $|\Psi|<\kappa$ . By the induction hypothesis, each $\psi(\bar{x},\bar{y})\in\Psi$ is equivalent to

\exists\bar{v}\bigwedge_{i<\lambda_{\psi}}\delta_{i,\psi}(\bar{v},\bar{x},\bar{y})=0,

where $\lambda_{\psi}\leqslant\kappa$ and

\delta_{i,\psi}(\bar{v},\bar{x},\bar{y})\doteq\sum_{\ell<\lambda_{\psi}}r_{i,\ell,\psi}v_{\ell}+\sum_{j<n}q_{i,j,\psi}x_{j}+\sum_{j<m}s_{i,j,\psi}y_{j},

with $r_{i,\ell,\psi},q_{i,j,\psi},s_{i,j,\psi}\in R$ and for each $i<\lambda$ , $r_{i,\ell,\psi}\neq 0$ for only finitely many $\ell<\lambda_{\psi}$ . After making the tuples $\bar{v}$ pairwise disjoint by tagging them with $\psi$ as $\bar{v}_{\psi}$ and setting $\lambda=\sum_{\psi\in\Psi}\lambda_{\psi}\leqslant\kappa$ , we obtain the required equivalent formula

\exists\bar{y}\exists(\bar{v}_{\psi})_{\psi\in\Psi}\bigwedge_{\psi\in\Psi}\bigwedge_{i<\lambda_{\psi}}\delta_{i,\psi}(\bar{v}_{\psi},\bar{x},\bar{y})=0.

∎

Proposition 2.6.

Let $\varphi(\bar{x})\in\Lambda_{\alpha}^{\kappa,\aleph_{0}}$ , where $\bar{x}=(x_{0},\ldots,x_{n-1})$ , $\kappa$ is a cardinal, and $\alpha$ is an ordinal. If $\kappa$ is regular, then $\varphi(\bar{x})$ is equivalent, relative to the theory of $R$ -modules, to a formula in $\Lambda_{\kappa}^{\kappa,\aleph_{0}}$ . If $\kappa$ is singular, then it is equivalent to a formula in $\Lambda_{\kappa^{+}}^{\kappa,\aleph_{0}}$ .

Proof.

We prove the regular case; the singular case is analogous. We show by induction on $\alpha\geqslant\kappa$ that every $\varphi(\bar{x})\in\Lambda_{\alpha}^{\kappa,\aleph_{0}}$ is equivalent to a formula in $\Lambda_{\kappa}^{\kappa,\aleph_{0}}$ .

If $\alpha=\kappa$ or $\alpha$ is a limit ordinal, there is nothing to prove. Assume that $\alpha=\beta+1$ . Then

\varphi(\bar{x})=\exists\bar{y}\bigwedge\{\psi(\bar{x},\bar{y})\mid\psi(\bar{x},\bar{y})\in\Psi\}

for some $|\bar{y}|=m<\aleph_{0}$ and some $\Psi\subseteq\Lambda_{\beta}^{\kappa,\aleph_{0}}$ with $|\Psi|<\kappa$ . By the induction hypothesis, each $\psi(\bar{x},\bar{y})\in\Psi$ is equivalent to a formula $\theta_{\psi}(\bar{x},\bar{y})\in\Lambda_{\kappa}^{\kappa,\aleph_{0}}$ . Choose $\alpha_{\psi}<\kappa$ such that $\theta_{\psi}(\bar{x},\bar{y})\in\Lambda_{\alpha_{\psi}}^{\kappa,\aleph_{0}}$ , and let $\gamma=\sup_{\psi\in\Psi}\alpha_{\psi}<\kappa$ , using the regularity of $\kappa$ . Then

\exists\bar{y}\bigwedge\{\theta_{\psi}(\bar{x},\bar{y})\mid\psi\in\Psi\}\in\Lambda_{\gamma+1}^{\kappa,\aleph_{0}}\subseteq\Lambda_{\kappa}^{\kappa,\aleph_{0}},

and this formula is equivalent to $\varphi(\bar{x})$ . ∎

Remark 2.7.

The arguments of Propositions 2.5 and 2.6 show that every formula in $\Lambda_{\infty}^{\aleph_{0},\aleph_{0}}$ is equivalent to a formula in $\Lambda_{1}^{\aleph_{0},\aleph_{0}}$ .

The following result will be very useful.

Proposition 2.8.

Let $\lambda\leqslant\kappa$ be cardinals, let $\alpha$ and $\epsilon$ be ordinals, and let $\varphi(\bar{x})\in\Lambda_{\alpha,\epsilon}^{\kappa,\lambda}$ .

(1)

If $A\models\varphi(\bar{a})$ for $\bar{a}\in A^{\epsilon}$ and $f:A\to B$ is an $R$ -homomorphism, then $B\models\varphi(f(\bar{a}))$ .
(2)

The set $\varphi(A)=\{\bar{a}\in A^{\epsilon}\mid A\models\varphi(\bar{a})\}$ is a subgroup of $A^{\epsilon}$ .

Proof.

We prove (1); the proof of (2) is analogous. Let $f:A\to B$ be an $R$ -homomorphism. We show by induction on $\alpha$ that, for every ordinal $\epsilon<\lambda$ and every $\varphi(\bar{x})\in\Lambda_{\alpha,\epsilon}^{\kappa,\lambda}$ , if $A\models\varphi(\bar{a})$ for $\bar{a}\in A^{\epsilon}$ , then $B\models\varphi(f(\bar{a}))$ .

If $\alpha=0$ , the result is clear because $\varphi(\bar{x})$ is a linear equation; if $\alpha$ is a limit ordinal, it follows immediately from the induction hypothesis. Assume that $\alpha=\beta+1$ and

\varphi(\bar{x})=\exists\bar{y}\bigwedge\{\psi(\bar{x},\bar{y})\mid\psi(\bar{x},\bar{y})\in\Psi\},

for some $|\bar{y}|=\zeta<\lambda$ and some $\Psi\subseteq\Lambda_{\beta,\epsilon+\zeta}^{\kappa,\lambda}$ with $|\Psi|<\kappa$ . Choose $\bar{c}\in A^{\zeta}$ such that $A\models\bigwedge\{\psi(\bar{a},\bar{c})\mid\psi(\bar{x},\bar{y})\in\Psi\}.$ By the induction hypothesis, $B\models\psi(f(\bar{a}),f(\bar{c}))$ for every $\psi\in\Psi$ . Hence $B\models\varphi(f(\bar{a}))$ . ∎

Definition 2.9.

Given $\Sigma\subseteq\Lambda_{\infty}^{\infty,\infty}$ , let $\leqslant_{\Sigma}$ be the binary relation given in $R\text{-}\mathrm{Mod}$ by $A\leqslant_{\Sigma}B$ if and only if $A\leqslant B$ and for every $\varphi(\bar{x})\in\Sigma$ and every $\bar{a}\in A^{|\bar{x}|}$ , $A\models\varphi(\bar{a})$ if and only if $B\models\varphi(\bar{a})$ .

Corollary 2.10.

Assume $\Sigma\subseteq\Lambda_{\infty}^{\infty,\infty}$ . If $A\leqslant_{\oplus}B$ , then $A\leqslant_{\Sigma}B$ .

Proof.

Since $A\leqslant_{\oplus}B$ , there is an $R$ -homomorphism $\pi:B\to A$ such that $\pi\restriction A=\mathrm{id}_{A}$ . The result now follows from Proposition 2.8. ∎

The following classical construction from module theory will be useful.

Remark 2.11.

The pushout of a pair of inclusions $A\leqslant B$ and $A\leqslant C$ in the category of $R$ -modules is a triple

(B\oplus_{A}C=(B\oplus C)/\widehat{A},\;f:B\to B\oplus_{A}C,\;g:C\to B\oplus_{A}C).

with $\widehat{A}=\{(a,-a)\mid a\in A\}$ , $f:b\mapsto[(b,0)]$ and $g:c\mapsto[(0,c)]$ .

Observe that $f,g$ are monomorphisms. Moreover, whenever $f^{\prime}:B\to D$ and $g^{\prime}:C\to D$ are $R$ -homomorphisms satisfying $f^{\prime}\restriction A=g^{\prime}\restriction A$ , the unique $R$ -homomorphism $t:B\oplus_{A}C\to D$ such that $t\circ f=f^{\prime}$ and $t\circ g=g^{\prime}$ is given by $t([(b,c)])=f^{\prime}(b)+g^{\prime}(c)$ .

2.2. Abstract elementary classes

Abstract elementary classes (AECs) were introduced by Shelah in the seventies [26]. We quickly introduce the notions of AECs used in this paper. For more details, see [1, §§4–8].

The following definition is due to Grossberg.

Definition 2.12.

Let $\mathbf{K}$ be a class of $L$ -structures and let $\preccurlyeq$ be a binary relation on $\mathbf{K}$ . We say that $(\mathbf{K},\preccurlyeq)$ is an abstract class if the following conditions are satisfied:

(1)

$\mathbf{K}$ and $\preccurlyeq$ are closed under isomorphisms, i.e., if $A\in\mathbf{K}$ and $f:A\to B$ is an isomorphism then $B\in\mathbf{K}$ and if $C\in\mathbf{K}$ is such that $C\preccurlyeq A$ then $f(C)\preccurlyeq B$ ;
(2)

if $A\preccurlyeq B$ , then $A$ is an $L$ -submodel of $B$ (written $A\leqslant B$ );
(3)

the relation $\preccurlyeq$ is a partial order on $\mathbf{K}$ .

Definition 2.13.

Let $(\mathbf{K},\preccurlyeq)$ be an abstract class, and let $A,B\in\mathbf{K}$ . An embedding $f:A\to B$ is a strong embedding (or $\preccurlyeq$ -embedding) if $f(A)\preccurlyeq B$ .

Definition 2.14.

Let $\mathbf{K}$ be a class of $L$ -structures and let $\preccurlyeq$ be a binary relation on $\mathbf{K}$ . We say that $(\mathbf{K},\preccurlyeq)$ is an abstract elementary class (AEC) if it is an abstract class, that is, if it satisfies Definition 2.12(1)–(3), and the following additional conditions hold:

(4)
Tarski–Vaught axioms: If $(A_{i})_{i<\delta}$ is an increasing continuous $\preccurlyeq$ -chain of structures in $\mathbf{K}$ for $\delta$ a limit ordinal, then:
1. (4.1)
  
  $\bigcup_{i<\delta}A_{i}\in\mathbf{K}$ ;
2. (4.2)
  
  for each $j<\delta$ , $A_{j}\preccurlyeq\bigcup_{i<\delta}A_{i}$ ;
3. (4.3)
  
  Smoothness axiom: if $A_{i}\preccurlyeq B$ for all $i<\delta$ , then $\bigcup_{i<\delta}A_{i}\preccurlyeq B$ .
(5)

Coherence axiom: If $A,B,C\in\mathbf{K}$ , $A\preccurlyeq C$ , $B\preccurlyeq C$ and $A\leqslant B$ , then $A\preccurlyeq B$ ;
(6)

Löwenheim–Skolem–Tarski axiom: There is a cardinal $\lambda\geqslant|L|+\aleph_{0}$ such that, whenever $B\in\mathbf{K}$ and $X\subseteq B$ , there is $A\in\mathbf{K}$ with $X\subseteq A\preccurlyeq B$ and $|A|\leqslant|X|+\lambda$ . The Löwenheim–Skolem number of $\mathcal{K}$ , denoted by $\mathrm{LS}(\mathcal{K})$ , is the least such cardinal $\lambda\geqslant|L|+\aleph_{0}$ .

When $(\mathbf{K},\preccurlyeq)$ is an AEC, we refer to $\preccurlyeq$ as a strong submodel relation on the class $\mathbf{K}$ .

Remark 2.15.

In this paper, $L$ is always the language of $R$ -modules, and $\mathbf{K}$ is usually the class of all $R$ -modules.

Throughout the rest of this subsection, $\mathcal{K}=(\mathbf{K},\preccurlyeq)$ is an AEC.

Definition 2.16.

$\mathcal{K}$ has

$\bullet$

the amalgamation property if every span $A\preccurlyeq B,C$ in $\mathcal{K}$ can be completed to a commutative square of strong embeddings;
$\bullet$

the joint embedding property if any two models in $\mathbf{K}$ strongly embed into a third model in $\mathbf{K}$ ;
$\bullet$

no maximal models if every model in $\mathbf{K}$ has a proper strong extension in $\mathbf{K}$ .

For $\lambda\geqslant\mathrm{LS}(\mathcal{K})$ , let $\mathbf{K}_{\lambda}=\{A\in\mathbf{K}\mid|A|=\lambda\}$ .

Definition 2.17.

Let $A\in\mathbf{K}$ , $X\subseteq A$ , and $\bar{b}$ be a sequence from $A$ . The (orbital or Galois) type of $\bar{b}$ over $X$ in $A$ , denoted by $\mathbf{gtp}_{\mathcal{K}}(\bar{b}/X;A)$ , is the equivalence class of $(\bar{b},X,A)$ modulo $E_{\mathcal{K}}$ , where $E_{\mathcal{K}}$ is the transitive closure of the relation $E_{\mathcal{K}}^{\mathrm{at}}$ defined as follows.

For $\ell\in\{1,2\}$ , $A_{\ell}\in\mathbf{K}$ , $\bar{b}_{\ell}\in A_{\ell}^{<\infty}$ , and $X_{\ell}\subseteq A_{\ell}$ , we set

(\bar{b}_{1},X_{1},A_{1})\mathrel{E_{\mathcal{K}}^{\mathrm{at}}}(\bar{b}_{2},X_{2},A_{2})

if $X:=X_{1}=X_{2}$ and there are $A_{*}\in\mathbf{K}$ and strong embeddings $f_{\ell}:A_{\ell}\to A_{*}$ , for $\ell\in\{1,2\}$ , such that $f_{1}\restriction X=f_{2}\restriction X=\mathrm{id}_{X}$ and $f_{1}(\bar{b}_{1})=f_{2}(\bar{b}_{2})$ .

Stability is a key model-theoretic dividing line that we study in this paper.

Definition 2.18.

(1)

If $A\in\mathbf{K}$ , let

\mathcal{S}_{\mathcal{K}}(A)=\{\mathbf{gtp}_{\mathcal{K}}(b/A;B)\mid A\preccurlyeq B\in\mathbf{K}\text{ and }b\in B\}.

(2)

Let $\lambda\geqslant\mathrm{LS}(\mathcal{K})$ . We say that $\mathcal{K}$ is $\lambda$ -stable if $|\mathcal{S}_{\mathcal{K}}(A)|\leqslant\lambda$ for every $A\in\mathbf{K}_{\lambda}$ .
(3)

$\mathcal{K}$ is stable if it is $\mu$ -stable for unboundedly many cardinals $\mu$ .

Given $p=\mathbf{gtp}_{\mathcal{K}}(b/A;B)\in\mathcal{S}_{\mathcal{K}}(A)$ and $X\subseteq A$ , let $p\restriction X=[(b,X,B)]_{E_{\mathcal{K}}}$ . The following notion was isolated in [12].

Definition 2.19.

$\mathcal{K}$ is $(<\aleph_{0})$ -tame if, for every $A\in\mathbf{K}$ and $p,q\in\mathcal{S}_{\mathcal{K}}(A)$ , if $p\neq q$ , then there is $X$ a finite subset of $A$ such that $p\restriction X\neq q\restriction X$ .

3. The lattice structure of $\mathscr{L}_{R}$

The main object of this paper is the following lattice.

Definition 3.1.

Let $R$ be a ring and let $\lambda\geqslant\operatorname{card}(R)+\aleph_{0}$ be a cardinal. Define $\mathscr{L}_{R,\lambda}$ to be the partially ordered class⁴⁴ 4 Corollary 3.5 shows that it is a lattice. of all binary relations $\preccurlyeq$ on $R\text{-}\mathrm{Mod}$ such that $(R\text{-}\mathrm{Mod},\preccurlyeq)$ is an AEC, $\mathrm{LS}(R\text{-}\mathrm{Mod},\preccurlyeq)\leqslant\lambda$ , and $\preccurlyeq$ refines the direct-summand relation. We order $\mathscr{L}_{R,\lambda}$ by reverse inclusion.⁵⁵ 5 Thus stronger strong submodel relations lie higher in the lattice. Let

\mathscr{L}_{R}=\bigcup_{\lambda\geqslant\operatorname{card}(R)+\aleph_{0}}\mathscr{L}_{R,\lambda}.

In this section, we establish basic properties of the lattices $\mathscr{L}_{R}$ and $\mathscr{L}_{R,\lambda}$ .

3.1. Basic results

Before we proceed it is important to notice that the structure of $\mathscr{L}_{R}$ and $\mathscr{L}_{R,\lambda}$ depends strongly on the ring $R$ . For instance, if $R$ is left semisimple, then $\mathscr{L}_{R}$ and $\mathscr{L}_{R,\lambda}$ consists only of the submodule relation.

Proposition 3.2.

Let $\lambda$ be a cardinal.

(1)

If $\{\preccurlyeq_{i}\mid i<\lambda\}\subseteq\mathscr{L}_{R,\lambda}$ , then $\bigcap_{i<\lambda}\preccurlyeq_{i}\in\mathscr{L}_{R,\lambda}$ .
(2)

If $\{\preccurlyeq_{i}\mid i<\theta\}\subseteq\mathscr{L}_{R}$ for some cardinal $\theta>0$ , then $\bigcap_{i<\theta}\preccurlyeq_{i}\in\mathscr{L}_{R}$ .

Proof.

We first prove (1). It is straightforward to verify all the AEC axioms for $(R\text{-}\mathrm{Mod},\bigcap_{i<\lambda}\preccurlyeq_{i})$ except the Löwenheim–Skolem–Tarski axiom. Let $X\subseteq B\in R\text{-}\mathrm{Mod}$ , and set $\mu=|X|+\lambda$ .

We recursively construct $\{A_{i,n}\mid i<\lambda,\ n<\omega\}$ inside $B$ , starting with $X\subseteq A_{0,0}$ , so that:

(1)

$|A_{i,n}|\leqslant\mu$ and $A_{i,n}\preccurlyeq_{i}B$ for every $i<\lambda$ and $n<\omega$ ;
(2)

$(A_{i,n})_{n<\omega}$ is an increasing $\preccurlyeq_{i}$ -chain for every $i<\lambda$ ;
(3)

$A_{i,n}\subseteq A_{j,n}$ whenever $i<j<\lambda$ and $n<\omega$ ;
(4)

$\bigcup_{i<\lambda}A_{i,n}\subseteq A_{0,n+1}$ for every $n<\omega$ .

The construction follows by recursion on $n$ , using the Löwenheim–Skolem–Tarski axiom in each $(R\text{-}\mathrm{Mod},\preccurlyeq_{i})$ inside $B$ and then the Coherence axiom.

Let $A=\bigcup_{n<\omega}A_{0,n}$ . Then $X\subseteq A$ and $|A|\leqslant\mu$ by Condition (1). Fix $i<\lambda$ . Conditions (3) and (4) give $A=\bigcup_{n<\omega}A_{i,n}$ , so Conditions (1) and (2) and the Tarski–Vaught axioms imply that $A\preccurlyeq_{i}B$ . Thus $A\,(\bigcap_{i<\lambda}\preccurlyeq_{i})\,B$ .

We now prove (2). The same construction works after replacing $\mu$ in Condition (1) by the size bound $\mu=|X|+\theta+\sup_{i<\theta}\mathrm{LS}(R\text{-}\mathrm{Mod},\preccurlyeq_{i}).$ Consequently, we have

\mathrm{LS}\left(R\text{-}\mathrm{Mod},\bigcap_{i<\theta}\preccurlyeq_{i}\right)\leqslant\theta+\sup_{i<\theta}\mathrm{LS}(R\text{-}\mathrm{Mod},\preccurlyeq_{i}),

and hence $\bigcap_{i<\theta}\preccurlyeq_{i}\in\mathscr{L}_{R}$ . ∎

Proposition 3.3.

Let $\lambda$ be a cardinal and $0<\theta$ be a cardinal (possibly finite).

(1)

If $\{\preccurlyeq_{i}\mid i<\theta\}\subseteq\mathscr{L}_{R,\lambda}$ , then this family has a greatest lower bound in $\mathscr{L}_{R,\lambda}$ .
(2)

If $\{\preccurlyeq_{i}\mid i<\theta\}\subseteq\mathscr{L}_{R}$ , then this family has a greatest lower bound in $\mathscr{L}_{R}$ .

Proof.

We prove (1); the proof of (2) is identical. Let

\Delta=\{\preccurlyeq\in\mathscr{L}_{R,\lambda}\mid\,\preccurlyeq_{i}\subseteq\preccurlyeq\text{ for every }i<\theta\}\text{ and }\preccurlyeq^{*}=\bigcap\Delta.

The set $\Delta$ is nonempty because the submodule relation belongs to it. It is straightforward to verify all the AEC axioms for $(R\text{-}\mathrm{Mod},\preccurlyeq^{*})$ except the Löwenheim–Skolem–Tarski axiom. Let $X\subseteq B\in R\text{-}\mathrm{Mod}$ . Applying the Löwenheim–Skolem–Tarski axiom to $X$ in $(R\text{-}\mathrm{Mod},\preccurlyeq_{0})$ inside $B$ , there is $A\preccurlyeq_{0}B$ such that $X\subseteq A$ and $|A|\leqslant|X|+\lambda$ . Since $\preccurlyeq_{0}\subseteq\preccurlyeq$ for every $\preccurlyeq\in\Delta$ , we have $A\preccurlyeq^{*}B$ . Hence $\preccurlyeq^{*}\in\mathscr{L}_{R,\lambda}$ , and it is straightforward to show that it is the greatest lower bound of the family.

For (2), define $\Delta$ inside $\mathscr{L}_{R}$ and use $\mathrm{LS}(R\text{-}\mathrm{Mod},\preccurlyeq_{0})$ in place of $\lambda$ . ∎

Remark 3.4.

In general, the relation $\preccurlyeq^{*}$ from Proposition 3.3 differs from $\bigcup_{i<\theta}\preccurlyeq_{i}$ . For example, let $R=\mathbb{Z}$ . Define $A\preccurlyeq_{1}B$ by requiring that divisibility by $2$ of elements of $A$ be preserved between $A$ and $B$ , and define $A\preccurlyeq_{2}B$ analogously using divisibility by $3$ . $\preccurlyeq_{1}\cup\preccurlyeq_{2}\neq\preccurlyeq^{*}$ because $(R\text{-}\mathrm{Mod},\preccurlyeq_{1}\cup\preccurlyeq_{2})$ fails to be an AEC since transitivity fails as witnessed by $6\mathbb{Z}\leqslant 3\mathbb{Z}\leqslant\mathbb{Z}$ .

Corollary 3.5.

For every ring $R$ and every cardinal $\lambda\geqslant\operatorname{card}(R)+\aleph_{0}$ , $\mathscr{L}_{R}$ and $\mathscr{L}_{R,\lambda}$ are lattices whose bottom element is the submodule relation.

Proof.

Proposition 3.2 provides joins (least upper bound), and Proposition 3.3 provides meets (greatest lower bound). ∎

The following result relies heavily on [2, Fact 3.1.1].

Lemma 3.6.

For every ring $R$ and every cardinal $\lambda\geqslant\operatorname{card}(R)+\aleph_{0}$ , $|\mathscr{L}_{R,\lambda}|\leqslant 2^{2^{\lambda}}$ .

Proof.

It is enough to show that $\mathscr{L}_{R,\lambda}^{=}:=\{\preccurlyeq\in\mathscr{L}_{R,\lambda}\mid\mathrm{LS}(R\text{-}\mathrm{Mod},\preccurlyeq)=\lambda\}$ is such that $|\mathscr{L}_{R,\lambda}^{=}|\leqslant 2^{2^{\lambda}}$ as $\mathscr{L}_{R,\lambda}=\bigcup_{\mu\leqslant\lambda}\mathscr{L}_{R,\mu}^{=}$ .

Let $L_{1}$ be the expansion of $\tau_{R}$ by $\{F_{i}^{n}(\bar{x})\mid i<\lambda,n<\omega\text{ and }|\bar{x}|=n\}$ . For each $\preccurlyeq\in\mathscr{L}_{R,\lambda}^{=}$ , the proof of Shelah’s Presentation Theorem yields a complete first-order $L_{1}$ -theory $T_{\preccurlyeq}$ and a set $\Gamma_{\preccurlyeq}$ of $L_{1}$ -types satisfying Conditions (1)–(3) of [2, Fact 3.1.1].

Let $\Phi:\mathscr{L}_{R,\lambda}^{=}\to\{T\mid T\text{ is a }L_{1}\text{-theory}\}\times\{\Gamma\mid\Gamma\text{ is a set of }L_{1}\text{-types}\}$ given by $\Phi(\preccurlyeq)=(T_{\preccurlyeq},\Gamma_{\preccurlyeq})$ . There are at most $2^{\lambda}$ possible $L_{1}$ -theories and at most $2^{2^{\lambda}}$ possible sets of $L_{1}$ -types, so it suffices to prove that $\Phi$ is injective.

Suppose $\Phi(\preccurlyeq)=\Phi(\preccurlyeq^{*})$ and $A\preccurlyeq B$ . By Condition (1) of [2, Fact 3.1.1], choose $A^{\prime}\in\operatorname{EC}(T_{\preccurlyeq},\Gamma_{\preccurlyeq})$ with $A^{\prime}\restriction\tau_{R}=A$ . By Condition (3) of [2, Fact 3.1.1], extend $A^{\prime}$ to $B^{\prime}\in\operatorname{EC}(T_{\preccurlyeq},\Gamma_{\preccurlyeq})$ such that $A^{\prime}\leqslant B^{\prime}$ as $L_{1}$ -structures and $B^{\prime}\restriction\tau_{R}=B$ . The equality of the two presentation pairs and another application of Condition (3) of [2, Fact 3.1.1], now for $\preccurlyeq^{*}$ , give $A\preccurlyeq^{*}B$ . Thus $\preccurlyeq\subseteq\preccurlyeq^{*}$ ; the reverse inclusion can be shown analogously. ∎

Proposition 3.7.

If $\preccurlyeq\,\in\,\mathscr{L}_{R}$ , then $(R\text{-}\mathrm{Mod},\preccurlyeq)$ has no maximal models and the joint embedding property.

Proof.

This follows by taking direct sums, since every direct-summand embedding is a $\preccurlyeq$ -embedding. ∎

3.2. When is $\leqslant_{\oplus}\in\mathscr{L}_{R}$ ?

We now characterize the rings $R$ for which $\leqslant_{\oplus}\in\mathscr{L}_{R}$ , equivalently, those for which $(R\text{-}\mathrm{Mod},\leqslant_{\oplus})$ is an AEC. The following proposition is immediate, so we omit its proof.

Proposition 3.8.

Let $B$ be an $R$ -module, let $\delta$ be a limit ordinal, and let $(A_{\alpha})_{\alpha<\delta}$ be a family of submodules of $B$ . If $\sum_{\alpha<\beta}A_{\alpha}$ is direct for every $\beta<\delta$ , then $\sum_{\alpha<\delta}A_{\alpha}$ is direct.

Lemma 3.9.

Assume that $(R\text{-}\mathrm{Mod},\leqslant_{\oplus})$ is an AEC with $\mathrm{LS}(R\text{-}\mathrm{Mod},\leqslant_{\oplus})=\kappa$ . Then every $R$ -module $B$ admits a decomposition $B=\bigoplus_{A\in\mathcal{X}}A$ for some family $\mathcal{X}$ of submodules of $B$ , each of cardinality at most $\kappa$ .

Proof.

Let $\mu=|B|$ , and fix an enumeration $(b_{\alpha})_{\alpha<\mu}$ of $B$ .

We recursively construct a sequence $(A_{\alpha})_{\alpha<\mu}$ of submodules of $B$ , each of cardinality at most $\kappa$ , such that for every $\alpha<\mu$ :

(1)

the sum $\sum_{\gamma<\alpha+1}A_{\gamma}$ is direct;
(2)

$\bigoplus_{\gamma<\alpha+1}A_{\gamma}$ is a direct summand of $B$ ;
(3)

$b_{\alpha}\in\bigoplus_{\gamma<\alpha+1}A_{\gamma}$ .

For $\alpha=0$ , apply the Löwenheim–Skolem–Tarski axiom to $b_{0}$ inside $B$ to obtain a direct summand $A_{0}$ of cardinality at most $\kappa$ containing $b_{0}$ .

Suppose $\alpha=\delta+1$ . By induction hypothesis, we have that $B=C\oplus D$ with $C=\bigoplus_{\gamma<\delta+1}A_{\gamma}.$ Let $\pi:B\to D$ be the projection given by the decomposition. Applying the Löwenheim–Skolem–Tarski axiom inside $D$ to $\pi(b_{\alpha})$ , choose a direct summand $A_{\alpha}$ of $D$ containing $\pi(b_{\alpha})$ and satisfying $|A_{\alpha}|\leqslant\kappa$ . Then $(A_{\gamma})_{\gamma<\alpha+1}$ has the required properties.

Suppose that $\alpha$ is a limit ordinal. By induction hypothesis, for any $\delta<\alpha$ , the sum $\sum_{\gamma<\delta}A_{\gamma}$ is direct and $\bigoplus_{\gamma<\delta}A_{\gamma}$ is a direct summand of $B$ . Now, by Proposition 3.8 we have that the sum $\sum_{\gamma<\alpha}A_{\gamma}$ is direct, and moreover:

\bigoplus_{\gamma<\alpha}A_{\gamma}=\bigcup_{\delta<\alpha}\bigoplus_{\gamma<\delta}A_{\gamma}.

Thus, by smoothness, this union is a direct summand of $B$ . We may therefore proceed exactly as in the successor case to choose $A_{\alpha}$ .

Finally, Conditions (1) and (3), together with Proposition 3.8, yield $B=\bigoplus_{\alpha<\mu}A_{\alpha}$ . Taking $\mathcal{X}=\{A_{\alpha}\mid\alpha<\mu\}$ completes the proof. ∎

Fact 3.10 ([23, Theorem 4.5.7]).

Let $R$ be a ring. The following are equivalent:

(1)

$R$ is left pure semisimple, i.e., $\leqslant_{\oplus}=\leqslant_{\mathrm{pp}}$ ;
(2)

There is a cardinal $\kappa$ such that every left $R$ -module is a direct sum of modules of cardinality less than $\kappa$ .

Theorem 3.11.

Let $R$ be a ring. $(R\text{-}\mathrm{Mod},\leqslant_{\oplus})$ is an AEC if and only if $R$ is left pure semisimple.

Proof.

The forward implication follows from Lemma 3.9 and Fact 3.10. Conversely, if $R$ is left pure semisimple, then $\leqslant_{\oplus}=\leqslant_{\mathrm{pp}}$ , and $(R\text{-}\mathrm{Mod},\leqslant_{\mathrm{pp}})$ is an AEC (see for example Proposition 4.4). ∎

If $\leqslant_{\oplus}\in\mathscr{L}_{R}$ , then $\mathscr{L}_{R}$ has a top element. This leaves open the following natural question.

Question 3.12.

Does the lattice $\mathscr{L}_{R}$ have a top element for every ring $R$ ? If the answer is negative, characterize the rings for which $\mathscr{L}_{R}$ has a top element.

4. The sublattices $\mathscr{L}^{1}_{R}$ and $\mathscr{L}^{2}_{R}$

We study the lattice $\mathscr{L}_{R}$ by dividing it into two natural sublattices: the one below purity and the one above purity.

4.1. The sublattice $\mathscr{L}^{1}_{R}$

We first study the sublattice given by the strong submodel relations on $R\text{-}\mathrm{Mod}$ which are weaker than the pure submodule relation.

Definition 4.1.

Given a ring $R$ , let $\mathscr{L}^{1}_{R}$ be the sublattice of $\mathscr{L}_{R}$ below the pure submodule relation.

The structure of $\mathscr{L}^{1}_{R}$ again depends strongly on $R$ . For instance, if $R$ is von Neumann regular, then purity coincides with the submodule relation [23, 2.3.22], so $\mathscr{L}^{1}_{R}$ has a single element. In this section, we show that, in general, $\mathscr{L}^{1}_{R}$ contains many well-behaved relations. In Section 5.1, we further investigate $\mathscr{L}^{1}_{\mathbb{Z}}$ .

Remark 4.2.

If $\preccurlyeq\in\mathscr{L}_{R}$ lies below $\leqslant_{\mathrm{pp}}$ , then $\mathrm{LS}(R\text{-}\mathrm{Mod},\preccurlyeq)=\operatorname{card}(R)+\aleph_{0}$ , because $\mathrm{LS}(R\text{-}\mathrm{Mod},\leqslant_{\mathrm{pp}})=\operatorname{card}(R)+\aleph_{0}$ . Thus, when studying the sublattice below $\leqslant_{\mathrm{pp}}$ , restricting from $\mathscr{L}_{R}$ to $\mathscr{L}_{R,\lambda}$ for any cardinal $\lambda$ does not change the class of relations.

Proposition 4.3.

$\mathscr{L}^{1}_{R}$ is a lattice with bottom element given by $\leqslant$ and top element given by $\leqslant_{\mathrm{pp}}$ . Moreover, $|\mathscr{L}^{1}_{R}|\leqslant 2^{2^{\operatorname{card}(R)+\aleph_{0}}}$ .

Proof.

This follows from Corollary 3.5. The moreover part follows from Remark 4.2 and Lemma 3.6. ∎

Proposition 4.4.

If $\Sigma$ is a set of first-order $\mathrm{pp}$ -formulas, then $\leqslant_{\Sigma}\in\mathscr{L}^{1}_{R}$ .

Proof.

This follows directly from the syntactic form of $\mathrm{pp}$ -formulas and the first-order downward Löwenheim–Skolem theorem. ∎

Example 4.5.

The following binary relations are in $\mathscr{L}^{1}_{R}$ :

(1)

$A$ is a submodule of $B$ .
(2)

$A$ is a pure submodule of $B$ .
(3)

$A$ is an RD-submodule of $B$ , i.e, $A\leqslant B$ and for every $r\in R$ , $rB\cap A=rA$ . We denote it by $A\leqslant_{\mathrm{RD}}B$ .

Definition 4.6.

A set $\Sigma$ of first-order $\mathrm{pp}$ -formulas is pleasant if:

(1)

every $\varphi(\bar{x})\in\Sigma$ has the form

$\exists\bar{y}\,(H\bar{y}=K\bar{x}),$

with $H\in\operatorname{Mat}_{m\times n}(R)$ , $\bar{y}=(y_{1},\ldots,y_{n})$ , $K\in\operatorname{Mat}_{m\times s}(R)$ , and $\bar{x}=(x_{1},\ldots,x_{s})$ ;
(2)

if $\exists\bar{y}\,(H\bar{y}=K\bar{x})\in\Sigma$ , then $\exists\bar{y}\,(H\bar{y}=\bar{z})\in\Sigma$ for $\bar{z}=(z_{1},\ldots,z_{m})$ .

A binary relation $\preccurlyeq$ on $R\text{-}\mathrm{Mod}$ is pleasant with respect to $\Sigma$ if $\Sigma$ is pleasant and $\preccurlyeq=\leqslant_{\Sigma}$ . We say $\preccurlyeq$ is pleasant if it is pleasant with respect to some such $\Sigma$ .

Example 4.7.

(1)

The submodule relation is pleasant with respect to $\Sigma=\emptyset$ .
(2)

The RD-submodule relation is pleasant with respect to

$\Sigma=\{\exists y\,(ry=x)\mid r\in R\}.$

(3)

The pure submodule relation is pleasant with respect to

\Sigma=\{\exists\bar{y}\,(H\bar{y}=K\bar{x})\mid H\in\operatorname{Mat}_{m\times n}(R),\ K\in\operatorname{Mat}_{m\times s}(R),\ m,n,s\in\mathbb{N}\}.

Remark 4.8.

Every pleasant relation belongs to $\mathscr{L}^{1}_{R}$ by Proposition 4.4.

Lemma 4.9.

If $\preccurlyeq$ is pleasant, then $(R\text{-}\mathrm{Mod},\preccurlyeq)$ has the amalgamation property.

Proof.

Let $A\preccurlyeq B,C$ , and form the pushout $(B\oplus_{A}C,f:B\to B\oplus_{A}C,g:C\to B\oplus_{A}C)$ with maps as in Remark 2.11.

Choose a pleasant set $\Sigma$ such that $\preccurlyeq=\leqslant_{\Sigma}$ . The maps $f$ and $g$ are monomorphisms, so it remains to show that the formulas in $\Sigma$ are reflected. We verify this for $f$ ; the proof for $g$ is analogous.

Suppose $B\oplus_{A}C\models\exists\bar{y}\,(H\bar{y}=Kf(\bar{\ell})),$ where $\exists\bar{y}\,(H\bar{y}=K\bar{x})\in\Sigma$ and $\bar{\ell}\in B^{s}$ . Choose $[(\bar{b},\bar{c})]\in(B\oplus_{A}C)^{n}$ witnessing this formula. Then there is $\bar{a}\in A^{m}$ such that

B\models H\bar{b}-K\bar{\ell}=\bar{a},\qquad C\models H\bar{c}=-\bar{a}.

As $\Sigma$ is pleasant, $\exists\bar{y}\,(H\bar{y}=\bar{z})\in\Sigma$ . Since $A\leqslant_{\Sigma}C$ , there is $\bar{e}\in A^{n}$ such that $A\models H\bar{e}=-\bar{a}$ . So $B\models H\bar{e}=-\bar{a}$ . Adding the last two equations in $B$ we have that $B\models H(\bar{b}+\bar{e})=K(\bar{\ell})$ . Therefore, $B\oplus_{A}C\models Hf(\bar{b}+\bar{e})=Kf(\bar{\ell})$ by Proposition 2.8. Hence $f[B]\leqslant_{\Sigma}B\oplus_{A}C$ . ∎

Definition 4.10.

Let $\Sigma$ be a set of $\mathrm{pp}$ -formulas, let $b\in B$ , and let $U\subseteq B$ . Define $\operatorname{pp}_{\Sigma}(b/U;B)$ to be

		$\displaystyle\{\varphi(z;\bar{u})\mid\varphi(z,\bar{x})\in\Sigma,\ \bar{u}\in U^{\|\bar{x}\|},\ B\models\varphi(b,\bar{u})\}$
		$\displaystyle\cup\{sz+ru=0\mid s,r\in R,\ u\in U,\ B\models sb+ru=0\}$
		$\displaystyle\cup\bigl\{\exists\bar{y}\,(H\bar{y}=\bar{u}+\bar{s}^{T}z)\,\bigm\|\,\exists\bar{y}\,(H\bar{y}=K\bar{x})\in\Sigma,\ \bar{u}\in U^{<\omega},\ \bar{s}\in R^{<\omega},$
		$\displaystyle B\models\exists\bar{y}\,(H\bar{y}=\bar{u}+\bar{s}^{T}b)\bigr\}.$

If $\Sigma$ is the set of all $\mathrm{pp}$ -formulas, we write $\operatorname{pp}(b/U;B)$ .

Remark 4.11.

The formulas on the second line and third line of the previous definition are themselves first-order pp-formulas. Therefore, when $\Sigma$ is the set of all pp-formulas, the last two lines do not add any additional information.

Theorem 4.12.

Assume that $\preccurlyeq$ is pleasant with respect to $\Sigma$ , and let $A\leqslant B,C$ . The following are equivalent:

(1)

$\mathbf{gtp}_{(R\text{-}\mathrm{Mod},\preccurlyeq)}(m/A;B)=\mathbf{gtp}_{(R\text{-}\mathrm{Mod},\preccurlyeq)}(n/A;C)$ ;
(2)

$\operatorname{pp}_{\Sigma}(m/A;B)=\operatorname{pp}_{\Sigma}(n/A;C)$ .

Proof.

Suppose that $\mathbf{gtp}_{(R\text{-}\mathrm{Mod},\preccurlyeq)}(m/A;B)=\mathbf{gtp}_{(R\text{-}\mathrm{Mod},\preccurlyeq)}(n/A;C)$ . By Lemma 4.9, equality of types may be witnessed in a common extension, so there are $D\in R\text{-}\mathrm{Mod}$ and $f:B\to D$ a $\preccurlyeq$ -embedding such that $C\preccurlyeq D$ , $f\upharpoonright A=\mathrm{id}_{A}$ and $f(m)=n$ . Since $\preccurlyeq=\leqslant_{\Sigma}$ , the formulas from the first line in the definition of $\operatorname{pp}_{\Sigma}$ agree for $m$ and $n$ . Since $f$ is a monomorphism, the same is true of the equations in the second line.

Suppose now that $B\models\exists\bar{y}\,(H\bar{y}=\bar{a}+\bar{s}^{T}m)$ and that $\exists\bar{y}\,(H\bar{y}=K\bar{x})\in\Sigma$ . Proposition 2.8 gives $D\models\exists\bar{y}\,(H\bar{y}=\bar{a}+\bar{s}^{T}n).$ Condition (2) of $\Sigma$ being pleasant and $C\leqslant_{\Sigma}D$ , imply that the latter formula already holds in $C$ . Thus every formula in $\operatorname{pp}_{\Sigma}(m/A;B)$ belongs to $\operatorname{pp}_{\Sigma}(n/A;C)$ ; the reverse inclusion is analogous.

Conversely, suppose that $\operatorname{pp}_{\Sigma}(m/A;B)=\operatorname{pp}_{\Sigma}(n/A;C)$ . Let $P=B\oplus_{A}C=(B\oplus C)/\widehat{A}$ with $\widehat{A}=\{(a,-a)\mid a\in A\}$ . Let $D\leqslant P$ be the cyclic submodule generated by $[(m,-n)]$ in $P$ . Let $Q=P/D$ and define $f(b)=[(b,0)]+D$ and $g(c)=[(0,c)]+D.$ Then $f\restriction A=g\restriction A$ and $f(m)=g(n)$ . We prove that $f$ is a $\preccurlyeq$ -embedding; the proof for $g$ is analogous.

First, we show that $f$ is a monormophism. If $f(b)=0$ , then there are $s\in R$ and $a\in A$ such that

b-sm=a\;\text{ and }\;sn=-a.

Thus $sz+a=0\in\operatorname{pp}_{\Sigma}(n/A;C)$ and hence $sz+a=0\in\operatorname{pp}_{\Sigma}(m/A;B)$ . Therefore $sm+a=0$ , and the first displayed equality gives $b=0$ .

It remains to show that $f[B]\leqslant_{\Sigma}Q$ . Suppose $Q\models\exists\bar{y}\,(H\bar{y}=Kf(\bar{\ell})),$ where $\exists\bar{y}\,(H\bar{y}=K\bar{x})\in\Sigma$ and $\bar{\ell}\in B^{s}$ . Choose $([(b_{1},c_{1})]+D,\ldots,[(b_{n},c_{n})]+D)\in Q^{n}$ witnessing the formula. Then there are $\bar{a}\in A^{m}$ and $\bar{s}\in R^{m}$ such that

B\models H\bar{b}-K\bar{\ell}-\bar{s}^{T}m=\bar{a},\qquad C\models H\bar{c}+\bar{s}^{T}n=-\bar{a}.

Hence $\exists\bar{y}\,(H\bar{y}=-\bar{a}-\bar{s}^{T}z)\in\operatorname{pp}_{\Sigma}(n/A;C)=\operatorname{pp}_{\Sigma}(m/A;B)$ . Choose $\bar{e}\in B^{n}$ such that $B\models H\bar{e}=-\bar{a}-\bar{s}^{T}m.$ Adding the last two equations in $B$ we have that $B\models H(\bar{b}+\bar{e})=K\bar{\ell}.$ Therefore, $Q\models Hf(\bar{b}+\bar{e})=Kf(\bar{\ell})$ by Proposition 2.8. Hence $f[B]\leqslant_{\Sigma}Q$ .∎

Remark 4.13.

When $\preccurlyeq=\leqslant_{\mathrm{pp}}$ and $\Sigma$ is the set from Example 4.7(3), the characterization of Theorem 4.12 already appears in [15, Lemma 3.14]. Nevertheless, the proof above is more direct as it avoids $\mathrm{pp}$ -quantifier elimination.

Remark 4.14.

The proof of Theorem 4.12 extends the methods used in [8, Theorem 2.16].

The preceding syntactic characterization of types yields tameness.

Corollary 4.15.

If $\preccurlyeq$ is pleasant, then $(R\text{-}\mathrm{Mod},\preccurlyeq)$ is $(<\aleph_{0})$ -tame.

Furthermore, we use Theorem 4.12 to obtain a direct proof of stability.

Proposition 4.16.

If $A$ is a left $R$ -module, then

|S_{\mathrm{pp}}(A)|\leqslant|A^{<\omega}|^{\operatorname{card}(R)+\aleph_{0}},

where $S_{\mathrm{pp}}(A)=\{\operatorname{pp}(m/A;B)\mid A\leqslant_{\mathrm{pp}}B,\ m\in B\}.$

Proof.

Fix a well-ordering of $A^{<\omega}$ , let $d\notin A^{<\omega}$ , and let $\Gamma$ be the set of all $\mathrm{pp}$ -formulas of the form $\varphi(z,\bar{x})$ .

Let $\Psi:S_{\mathrm{pp}}(A)\longrightarrow(A^{<\omega}\cup\{d\})^{\Gamma}$ be given by $\Psi(\mathrm{pp}(m/A,B)):\Gamma\to A^{<\omega}\cup\{d\}$ where for $\varphi(z,\bar{x})\in\Gamma$ , let $\Psi(\mathrm{pp}(m/A,B))(\varphi)$ be the least tuple $\bar{a}\in A^{|\bar{x}|}$ such that $B\models\varphi(m,\bar{a})$ , if such a tuple exists, and let $\Psi(\mathrm{pp}(m/A,B))(\varphi)=d$ otherwise.

We show that $\Psi$ is injective. Suppose $\Psi(\operatorname{pp}(m_{1}/A;B_{1}))=\Psi(\operatorname{pp}(m_{2}/A;B_{2})).$ It suffices to prove one inclusion. Let $\varphi(z,\bar{a})\in\operatorname{pp}(m_{1}/A;B_{1})$ , and write $\bar{a}_{\varphi}=\Psi(\operatorname{pp}(m_{1}/A;B_{1}))(\varphi)$ . Then $B_{1}\models\exists z\bigl(\varphi(z,\bar{a})\wedge\varphi(z,\bar{a}_{\varphi})\bigr).$ Since $A\leqslant_{\mathrm{pp}}B_{1},B_{2}$ and the previous formula is a first-order $\mathrm{pp}$ -formula, $B_{2}\models\exists z\bigl(\varphi(z,\bar{a})\wedge\varphi(z,\bar{a}_{\varphi})\bigr).$ As solution sets of $\mathrm{pp}$ -formulas are cosets [22, Corollary 2.2], hence $\varphi(B_{2},\bar{a})=\varphi(B_{2},\bar{a}_{\varphi})$ . Since $\bar{a}_{\varphi}=\Psi(\mathrm{pp}(m_{1}/A,B_{1}))(\varphi)=\Psi(\mathrm{pp}(m_{2}/A,B_{2}))(\varphi)$ , $B_{2}\models\varphi(m_{2},\bar{a}_{\varphi})$ , and therefore $B_{2}\models\varphi(m_{2},\bar{a})$ .

Finally, $|\Gamma|\leqslant\operatorname{card}(R)+\aleph_{0}$ as $\Gamma$ is a set of first-order formulas, which gives the stated bound. ∎

Lemma 4.17.

Assume that $\preccurlyeq$ is pleasant. If $\lambda^{\operatorname{card}(R)+\aleph_{0}}=\lambda$ , then $(R\text{-}\mathrm{Mod},\preccurlyeq)$ is $\lambda$ -stable.

Proof.

Let $A\in(R\text{-}\mathrm{Mod})_{\lambda}$ and $\preccurlyeq$ be pleasant with respect to $\Sigma$ . Suppose, toward a contradiction, that $\{p_{i}=\mathbf{gtp}_{(R\text{-}\mathrm{Mod},\preccurlyeq)}(m_{i}/A;B)\mid i<\lambda^{+}\}$ is a family of distinct types in $\mathcal{S}_{(R\text{-}\mathrm{Mod},\preccurlyeq)}(A)$ . By amalgamation, we may assume that all the types are realized in the same module $B$ .

Choose $\widehat{A}\leqslant_{\mathrm{pp}}B$ such that $A\subseteq\widehat{A}$ and $|\widehat{A}|\leqslant\lambda$ . Proposition 4.16 and the hypothesis on $\lambda$ give that

|S_{\mathrm{pp}}(\widehat{A})|\leqslant|\widehat{A}^{<\omega}|^{\operatorname{card}(R)+\aleph_{0}}\leqslant\lambda.

Hence there are distinct $i,j<\lambda^{+}$ with $\operatorname{pp}(m_{i}/\widehat{A};B)=\operatorname{pp}(m_{j}/\widehat{A};B).$ Hence $\mathrm{pp}_{\Sigma}(m_{i}/A,B)=\mathrm{pp}_{\Sigma}(m_{j}/A,B)$ as the formulas that appear in these types are pp-formulas. Therefore, we have that

\mathbf{gtp}_{(R\text{-}\mathrm{Mod},\preccurlyeq)}(m_{i}/A,B)=\mathbf{gtp}_{(R\text{-}\mathrm{Mod},\preccurlyeq)}(m_{j}/A,B)

by Theorem 4.12. This is a contradiction to $p_{i}\neq p_{j}$ . ∎

Remark 4.18.

It follows from Lemma 4.17 and Example 4.7 that the AEC $(R\text{-}\mathrm{Mod},\preccurlyeq)$ is $\lambda$ -stable whenever $\lambda^{\operatorname{card}(R)+\aleph_{0}}=\lambda$ and $\preccurlyeq\in\{\leqslant,\leqslant_{\mathrm{RD}},\leqslant_{\mathrm{pp}}\}$ . These three results were first obtained in [17, Lemma 3.6], [19, Theorem 3.11], and [15, Theorem 3.16], respectively.

Question 4.19.

Let $\Sigma$ be a set of first-order $\mathrm{pp}$ -formulas. Do Lemma 4.9, Corollary 4.15, and Lemma 4.17 hold for $(R\text{-}\mathrm{Mod},\leqslant_{\Sigma})$ ?

4.2. The sublattice $\mathscr{L}^{2}_{R}$

We now study the sublattice given by the strong submodel relations on $R\text{-}\mathrm{Mod}$ which are stronger than the pure submodule relation.

Definition 4.20.

Given a ring $R$ , let $\mathscr{L}^{2}_{R}$ be the sublattice of $\mathscr{L}_{R}$ above the pure submodule relation.

The structure of $\mathscr{L}^{2}_{R}$ again depends strongly on the ring $R$ . For instance, if $R$ is left pure semisimple, then purity coincides with the direct summand relation [23, Theorem 4.5.7], so $\mathscr{L}^{2}_{R}$ has a single element. In this section, we exhibit points of $\mathscr{L}^{2}_{R}$ and show that many of them have good stability-theoretic behavior. In Sections 5.2 and 5.3, we further investigate $\mathscr{L}_{\mathbb{Z}}$ .

Proposition 4.21.

$\mathscr{L}^{2}_{R}$ is a lattice with bottom element given by $\leqslant_{\mathrm{pp}}$

Our main tool for finding new points in $\mathscr{L}^{2}_{R}$ is the use of infinitary $\mathrm{pp}$ -formulas.

Definition 4.22.

Let $\lambda\leqslant\kappa$ be infinite cardinals, let $\alpha$ be an ordinal, and let $A,B\in R\text{-}\mathrm{Mod}$ . We say that $A$ is $(\kappa,\lambda,\alpha)$ -pure in $B$ , denoted by $A\leqslant^{(\kappa,\lambda,\alpha)}_{\mathrm{pp}}B$ , if $A\leqslant_{\Lambda^{\kappa,\lambda}_{\alpha}}B$ . Equivalently, $A\leqslant B$ and, for every $\varphi(\bar{x})\in\Lambda^{\kappa,\lambda}_{\alpha}$ and every $\bar{a}\in A^{|\bar{x}|}$ ,

A\models\varphi(\bar{a})\quad\Longleftrightarrow\quad B\models\varphi(\bar{a}).

When $\lambda=\aleph_{0}$ , we write $\leqslant^{(\kappa,\alpha)}_{\mathrm{pp}}$ instead of $\leqslant^{(\kappa,\aleph_{0},\alpha)}_{\mathrm{pp}}$ .

Similarly, we write $A\leqslant^{(\kappa,\lambda,\infty)}_{\mathrm{pp}}B$ if $A\leqslant_{\Lambda^{\kappa,\lambda}_{\infty}}B$ . When $\lambda=\aleph_{0}$ , we write $\leqslant^{(\kappa,\infty)}_{\mathrm{pp}}$ instead of $\leqslant^{(\kappa,\aleph_{0},\infty)}_{\mathrm{pp}}$ .

Proposition 4.23.

(1)

$A\leqslant_{\mathrm{pp}}B$ if and only if $A\leqslant^{(\aleph_{0},\aleph_{0},\omega)}_{\mathrm{pp}}B$ if and only if $A\leqslant^{(\aleph_{0},\aleph_{0},1)}_{\mathrm{pp}}B$ .
(2)

For every regular cardinal $\kappa$ , $A\leqslant^{(\kappa,\infty)}_{\mathrm{pp}}B$ if and only if $A\leqslant^{(\kappa,\kappa)}_{\mathrm{pp}}B$ .
(3)

For every singular cardinal $\kappa$ , $A\leqslant^{(\kappa,\infty)}_{\mathrm{pp}}B$ if and only if $A\leqslant^{(\kappa,\kappa^{+})}_{\mathrm{pp}}B$ .

Proof.

Part (1) follows from Remark 2.7, while parts (2) and (3) follow from Proposition 2.6. ∎

Remark 4.24.

By Corollary 2.10, every relation introduced in Definition 4.22 refines the direct summand relation.

For a cardinal $\theta$ , let $\theta^{-}:=\theta$ if $\theta$ is a limit cardinal, and let $\theta^{-}:=\mu$ if $\theta=\mu^{+}$ .

Lemma 4.25.

For every cardinal $\kappa$ and every ordinal $\alpha$ , $(R\text{-}\mathrm{Mod},\leqslant^{(\kappa,\alpha)}_{\mathrm{pp}})$ is an AEC and

\mathrm{LS}(R\text{-}\mathrm{Mod},\leqslant^{(\kappa,\alpha)}_{\mathrm{pp}})\leqslant 2^{(\kappa+\operatorname{card}(R)+\aleph_{0})^{-}}.

Moreover, $(R\text{-}\mathrm{Mod},\leqslant^{(\kappa,\infty)}_{\mathrm{pp}})$ is an AEC and $\mathrm{LS}(R\text{-}\mathrm{Mod},\leqslant^{(\kappa,\infty)}_{\mathrm{pp}})\leqslant 2^{(\kappa+\operatorname{card}(R)+\aleph_{0})^{-}}.$

Proof.

The moreover part follows from Proposition 4.23(2) and (3), so it is enough to prove the main assertion. Fix a cardinal $\kappa$ and an ordinal $\alpha$ . It is clear that $(R\text{-}\mathrm{Mod},\leqslant^{(\kappa,\alpha)}_{\mathrm{pp}})$ is an abstract class. We verify Conditions (4)–(6) of Definition 2.14.

(4)

Let $(A_{i})_{i<\delta}$ be an increasing continuous $\leqslant^{(\kappa,\alpha)}_{\mathrm{pp}}$ -chain, and let $A:=\bigcup_{i<\delta}A_{i}$ . Condition (4.1) is clear. We prove Condition (4.2); the proof of Condition (4.3) is similar.

We show by induction on $\beta\leqslant\alpha$ that, for every $i<\delta$ , every $\varphi(\bar{x})\in\Lambda^{\kappa,\aleph_{0}}_{\beta}$ , and every $\bar{a}\in A_{i}^{|\bar{x}|}$ , $A\models\varphi(\bar{a})$ if and only if $A_{i}\models\varphi(\bar{a}).$

If $\beta=0$ , the assertion follows because every formula in $\Lambda^{\kappa,\aleph_{0}}_{0}$ is a linear equation. If $\beta$ is a limit ordinal, the assertion follows directly from the induction hypothesis. Suppose that $\beta=\gamma+1$ , fix $i<\delta$ , let $\varphi(\bar{x})\in\Lambda^{\kappa,\aleph_{0}}_{\gamma+1}$ , and let $\bar{a}\in A_{i}^{|\bar{x}|}$ .

By Proposition 2.8, $A_{i}\models\varphi(\bar{a})$ implies $A\models\varphi(\bar{a})$ . Conversely, suppose that $A\models\varphi(\bar{a})$ . Then

$\varphi(\bar{x})=\exists\bar{y}\bigwedge_{\psi\in\Psi}\psi(\bar{x},\bar{y}),$

for some $|\bar{y}|<\aleph_{0}$ and some $\Psi\subseteq\Lambda^{\kappa,\aleph_{0}}_{\gamma}$ with $|\Psi|<\kappa$ . Choose a finite tuple $\bar{b}\in A$ such that $A\models\bigwedge_{\psi\in\Psi}\psi(\bar{a},\bar{b}).$ There is $j\geqslant i$ such that $\bar{a},\bar{b}\in A_{j}$ . By the induction hypothesis, $A_{j}\models\bigwedge_{\psi\in\Psi}\psi(\bar{a},\bar{b}),$ so $A_{j}\models\varphi(\bar{a})$ . Since $A_{i}\leqslant^{(\kappa,\alpha)}_{\mathrm{pp}}A_{j}$ , we conclude that $A_{i}\models\varphi(\bar{a})$ .
(5)

Coherence is immediate.
(6)

Since $\Lambda^{\kappa,\aleph_{0}}_{\alpha}\subseteq\mathfrak{L}_{\kappa,\omega}(\tau_{R})$ , the downward Löwenheim–Skolem theorem for $\mathfrak{L}_{\kappa,\omega}(\tau_{R})$ yields the desired inequality (se for example [16, Theorem 1.23]).

∎

We next introduce a point of $\mathscr{L}^{2}_{R}$ that admits both an infinitary $\mathrm{pp}$ -characterization and an algebraic characterization. In Section 5.3, we show that, over $\mathbb{Z}$ , it acts as a dividing point for the amalgamation property.

Definition 4.26.

A weak infinitary $\mathrm{pp}$ -formula is a formula of the form

\exists\bar{y}\bigwedge_{i<\kappa}\psi_{i}(\bar{x},\bar{y}),

with $|\bar{x}|,|\bar{y}|<\aleph_{0}$ , $\kappa$ is a cardinal, and each $\psi_{i}(\bar{x},\bar{y})$ is a first-order $\mathrm{pp}$ -formula. We denote the set of all weak $\mathrm{pp}$ -formulas by $\Lambda^{\mathrm{wpp}}_{\infty}$ .

For $A,B\in R\text{-}\mathrm{Mod}$ , we define $A\leqslant^{\mathrm{weak}}_{\mathrm{pp,\infty}}B$ if and only if $A\leqslant_{\Lambda^{\mathrm{wpp}}_{\infty}}B$ . Equivalently, $A\leqslant B$ and, for every $\varphi(\bar{x})\in\Lambda^{\mathrm{wpp}}_{\infty}$ and every $\bar{a}\in A^{|\bar{x}|}$ ,

A\models\varphi(\bar{a})\quad\Longleftrightarrow\quad B\models\varphi(\bar{a}).

Remark 4.27.

Suppose that there is a finitely generated $R$ -module $A$ which is not pure injective. Then $\leqslant^{\mathrm{weak}}_{\mathrm{pp,\infty}}$ is strictly stronger than $\leqslant_{\mathrm{pp}}$ .

Indeed, let $B$ be the pure injective envelope of $A$ and fix a finite generating tuple $\bar{a}$ for $A$ . We have that $A\leqslant_{\mathrm{pp}}B$ and $A\lneq B$ . Since $A$ is not pure-injective, there is a $\mathrm{pp}$ -type over $A$ which is finitely satisfiable in $A$ but not realized in $A$ (see for example [22, Theorem 2.8]). As every parameter from $A$ is an $R$ -linear combination of $\bar{a}$ , we may write this type as $p(y,\bar{a})$ . Since $B$ is pure-injective, $p(y,\bar{a})$ is realized in $B$ . Hence

B\models\exists y\bigwedge_{\varphi\in p}\varphi(y,\bar{a})\quad\text{but}\quad A\not\models\exists y\bigwedge_{\varphi\in p}\varphi(y,\bar{a}).

The displayed formula is a weak infinitary $\mathrm{pp}$ -formula, so $A\not\leqslant^{\mathrm{weak}}_{\mathrm{pp,\infty}}B$ .

In particular, the hypothesis holds for $R=\mathbb{Z}$ , taking $A=\mathbb{Z}$ .

Remark 4.28.

There are at most $\operatorname{card}(R)+\aleph_{0}$ first-order $\mathrm{pp}$ -formulas in a finite number of variables $\bar{x},\bar{y}$ . Hence every $\varphi(\bar{x})\in\Lambda^{\mathrm{wpp}}_{\infty}$ is equivalent, modulo the theory of $R$ -modules, to a formula in $\Lambda^{(\operatorname{card}(R)+\aleph_{0})^{+},\aleph_{0}}_{2}$ . In particular, $\leqslant^{((\operatorname{card}(R)+\aleph_{0})^{+},\infty)}_{\mathrm{pp}}\subseteq\leqslant^{\mathrm{weak}}_{\mathrm{pp,\infty}},$ and $\leqslant^{\mathrm{weak}}_{\mathrm{pp,\infty}}$ refines the direct summand relation.

Lemma 4.29.

The class $(R\text{-}\mathrm{Mod},\leqslant^{\mathrm{weak}}_{\mathrm{pp,\infty}})$ is an AEC with

\mathrm{LS}(R\text{-}\mathrm{Mod},\leqslant^{\mathrm{weak}}_{\mathrm{pp,\infty}})\leqslant 2^{\operatorname{card}(R)+\aleph_{0}}.

Proof.

The proof is similar to Lemma 4.25 where $\mathrm{LS}(R\text{-}\mathrm{Mod},\leqslant^{\mathrm{weak}}_{\mathrm{pp,\infty}})\leqslant 2^{\mathrm{card}(R)+\aleph_{0}}$ follows from Remark 4.28. ∎

We use the following definition from [31].

Definition 4.30.

Let $A$ and $B$ be modules, and let $Z\subseteq A$ . A function $f:Z\to B$ is a $\mathrm{pp}$ - $(A,B)$ -homomorphism if, for every first-order $\mathrm{pp}$ -formula $\varphi(\bar{x})$ and every $\bar{z}\in Z^{|\bar{x}|}$ ,

A\models\varphi(\bar{z})\quad\Longrightarrow\quad B\models\varphi(f(\bar{z})).

Lemma 4.31.

Let $A,B\in R\text{-}\mathrm{Mod}$ . The following are equivalent:

(1)

$A\leqslant^{\mathrm{weak}}_{\mathrm{pp,\infty}}B$ .
(2)

$A\leqslant B$ and, for every $\bar{a}\in A^{<\omega}$ and $\bar{b}\in B^{<\omega}$ , there is a $\mathrm{pp}$ - $(B,A)$ -homomorphism $f$ whose domain contains the entries of $\bar{a}\bar{b}$ , such that $f(\bar{a})=\bar{a}$ and $f(\bar{b})\in A^{|\bar{b}|}$ .

Proof.

Suppose first that $A\leqslant^{\mathrm{weak}}_{\mathrm{pp,\infty}}B$ . Fix $\bar{a}\in A^{<\omega}$ and $\bar{b}\in B^{<\omega}$ , and let $p=\operatorname{pp}(\bar{b}/\bar{a};B)$ . Then $B\models\bigwedge_{\varphi\in p}\varphi(\bar{b},\bar{a}),$ and hence $B\models\exists\bar{y}\bigwedge_{\varphi\in p}\varphi(\bar{y},\bar{a}).$ Since $A\leqslant^{\mathrm{weak}}_{\mathrm{pp,\infty}}B$ , there is $\bar{c}\in A^{|\bar{b}|}$ such that $A\models\bigwedge_{\varphi\in p}\varphi(\bar{c},\bar{a}).$ The function on the entries of $\bar{a}\bar{b}$ that fixes $\bar{a}$ and sends $\bar{b}$ to $\bar{c}$ is therefore a $\mathrm{pp}$ - $(B,A)$ -homomorphism.

Conversely, let

\varphi(\bar{x})=\exists\bar{y}\bigwedge_{i<\kappa}\psi_{i}(\bar{x},\bar{y}),

let $\bar{a}\in A^{|\bar{x}|}$ , and suppose that $B\models\varphi(\bar{a})$ . Choose $\bar{b}\in B^{|\bar{y}|}$ such that $B\models\bigwedge_{i<\kappa}\psi_{i}(\bar{a},\bar{b}).$ By assumption, there is a $\mathrm{pp}$ - $(B,A)$ -homomorphism $f$ that fixes $\bar{a}$ and sends $\bar{b}$ into $A$ . Thus $A\models\bigwedge_{i<\kappa}\psi_{i}(\bar{a},f(\bar{b})),$ so $A\models\varphi(\bar{a})$ . The reverse implication follows from Proposition 2.8, and hence $A\leqslant^{\mathrm{weak}}_{\mathrm{pp,\infty}}B$ . ∎

We now show that stability is pervasive in $\mathscr{L}^{2}_{R}$ . This contrasts with amalgamation, which, as we will show in Section 5.3, often fails.

Definition 4.32.

A set $\Sigma\subseteq\Lambda^{\lambda^{+},\lambda^{+}}_{1}$ , where $\lambda$ is an infinite cardinal with $\operatorname{card}(R)\leqslant\lambda$ , is E-nice if every $\varphi(\bar{x})\in\Sigma$ has the form

\exists\bar{v}\bigwedge_{i<\lambda}\delta_{i}(\bar{v},\bar{x})=0,

with $\bar{x}=(x_{0},\ldots,x_{n-1})$ , $\bar{v}=(v_{\ell})_{\ell<\lambda}$ and

\delta_{i}(\bar{v},\bar{x})\doteq\sum_{\ell<\lambda}r_{i,\ell}v_{\ell}+\sum_{j<n}q_{i,j}x_{j},

with $r_{i,\ell},q_{i,j}\in R$ and for each $i<\lambda$ , $r_{i,\ell}\neq 0$ for only finitely many $\ell<\lambda$ .

A binary relation $\preccurlyeq$ on $R\text{-}\mathrm{Mod}$ is E-nice with respect to $\Sigma$ if $\Sigma$ is E-nice and $\preccurlyeq=\leqslant_{\Sigma}$ . We say $\preccurlyeq$ is E-nice if it is E-nice with respect to some such $\Sigma$ .

Remark 4.33.

$\bullet$

It is important to note that if $\preccurlyeq$ is E-nice, we do not necessarily have that $\preccurlyeq\in\mathscr{L}_{R}$ .
$\bullet$

If $\Sigma\subseteq\Lambda^{\lambda^{+},\lambda^{+}}_{1}$ is E-nice, then $|\Sigma|\leqslant 2^{\lambda}$ by the syntactic structure of the formulas in $\Sigma$ .

Lemma 4.34.

If $\,\preccurlyeq\in\{\leqslant^{(\kappa,\alpha)}_{\mathrm{pp}}\mid\kappa\in\mathrm{Card},\ \alpha\in\mathrm{Ord}\}\cup\{\leqslant^{(\kappa,\infty)}_{\mathrm{pp}}\mid\kappa\in\mathrm{Card}\}\cup\{\leqslant^{\mathrm{weak}}_{\mathrm{pp,\infty}}\},$ then $\preccurlyeq$ is E-nice.

Proof.

For $\preccurlyeq\,\in\{\leqslant^{(\kappa,\alpha)}_{\mathrm{pp}}\,\mid\,\kappa\in\mathrm{Card}\,,\alpha\in\mathrm{Ord}\}$ it follows from Proposition 2.5. For $\preccurlyeq\,\in\{\leqslant^{(\kappa,\infty)}_{\mathrm{pp}}\,\mid\,\kappa\in\mathrm{Card}\}$ it follows from Propositions 4.23 and 2.5. For $\preccurlyeq=\leqslant^{\mathrm{weak}}_{\mathrm{pp,\infty}}$ it follows from Remark 4.28 and Proposition 2.5. ∎

Definition 4.35.

Suppose that

\varphi(\bar{x}):=\exists\bar{v}\bigwedge_{i<\lambda}\delta_{i}(\bar{v},\bar{x})=0,

with $\bar{x}=(x_{0},\ldots,x_{n-1})$ , $\bar{v}=(v_{\ell})_{\ell<\lambda}$ and

\delta_{i}(\bar{v},\bar{x})\doteq\sum_{\ell<\lambda}r_{i,\ell}v_{\ell}+\sum_{j<n}q_{i,j}x_{j}.

with $r_{i,\ell},q_{i,j}\in R$ and for each $i<\lambda$ , $r_{i,\ell}\neq 0$ for only finitely many $\ell<\lambda$ .

Let $\varphi^{*}(\bar{x}^{\prime},\bar{x}^{\prime\prime})$ be the formula

\exists\bar{v}\bigwedge_{i<\lambda}\delta_{i}^{*}(\bar{v},\bar{x}^{\prime},\bar{x}^{\prime\prime})=0,

with $\bar{x}^{\prime}=(x_{0},\ldots,x_{n-1})$ , $\bar{x}^{\prime\prime}=(x_{n},\ldots,x_{2n-1})$ , and

\delta_{i}^{*}(\bar{v},\bar{x}^{\prime},\bar{x}^{\prime\prime})\doteq\sum_{\ell<\lambda}r_{i,\ell}v_{\ell}+\sum_{j<n}q_{i,j}x_{j}+\sum_{j<n}q_{i,j}x_{n+j}.

Proposition 4.36.

Assume that $\preccurlyeq$ is E-nice with respect to $\Sigma$ and that $\mathcal{K}=(\mathbf{K},\preccurlyeq)$ is an AEC with $\mathbf{K}\subseteq R\text{-}\mathrm{Mod}$ . If $\varphi(\bar{x})\in\Sigma$ , $A\preccurlyeq B$ with $A,B\in\mathbf{K}$ , and $\bar{a},\bar{a}^{\prime}\in A^{|\bar{x}|}$ , then $A\models\varphi^{*}(\bar{a},\bar{a}^{\prime})$ if and only if $B\models\varphi^{*}(\bar{a},\bar{a}^{\prime}).$

Proof.

The forward implication follows from Proposition 2.8. Conversely, suppose that $B\models\varphi^{*}(\bar{a},\bar{a}^{\prime})$ . Then $B\models\varphi(\bar{a}+\bar{a}^{\prime})$ . Since $\bar{a}+\bar{a}^{\prime}\in A^{|\bar{x}|}$ , $\varphi(\bar{x})\in\Sigma$ , and $A\leqslant_{\Sigma}B$ , we have that $A\models\varphi(\bar{a}+\bar{a}^{\prime})$ . Hence $A\models\varphi^{*}(\bar{a},\bar{a}^{\prime})$ . ∎

If $A,C\leqslant B$ , let $\langle AC\rangle_{B}$ denote the submodule of $B$ generated by $A\cup C$ . Thus $\langle AC\rangle_{B}=A+C=\{a+c\mid a\in A,\ c\in C\}.$

Remark 4.37.

Assume that $\preccurlyeq$ is E-nice with respect to $\Sigma$ and that $\mathcal{K}=(\mathbf{K},\preccurlyeq)$ is an AEC with $\mathbf{K}\subseteq R\text{-}\mathrm{Mod}$ . Suppose that $A,C\leqslant B$ , that $\langle AC\rangle_{B},B\in\mathbf{K}$ , and that $\langle AC\rangle_{B}\not\preccurlyeq B$ . Then there are $\varphi(\bar{x})\in\Sigma$ , with $|\bar{x}|=n<\omega$ , and $\bar{d}\in\langle AC\rangle_{B}^{n}$ such that

B\models\varphi(\bar{d})\quad\text{and}\quad\langle AC\rangle_{B}\not\models\varphi(\bar{d}).

Writing $\bar{d}=\bar{a}+\bar{c}$ , with $\bar{a}\in A^{n}$ and $\bar{c}\in C^{n}$ , we obtain

B\models\varphi^{*}(\bar{a},\bar{c})\quad\text{and}\quad\langle AC\rangle_{B}\not\models\varphi^{*}(\bar{a},\bar{c}).

Theorem 4.38.

Assume that $\preccurlyeq$ is E-nice. If $\mathbf{K}\subseteq R\text{-}\mathrm{Mod}$ satisfies the following conditions:

(1)

$\mathcal{K}=(\mathbf{K},\preccurlyeq)$ is an AEC;
(2)

$\mathbf{K}$ is closed under $R$ -submodules, i.e., if $A\leqslant B$ and $B\in\mathbf{K}$ , then $A\in\mathbf{K}$ .

Then $\mathcal{K}$ is stable.

Proof.

Let $\Sigma\subseteq\Lambda^{\lambda^{+},\lambda^{+}}_{1}$ E-nice for $\lambda$ a cardinal such that $\preccurlyeq=\leqslant_{\Sigma}$ . A standard witness closing construction, using that $|\Sigma|\leqslant 2^{\lambda}$ from Remark 4.33 and the closure of $\mathbf{K}$ under submodules, gives $\mathrm{LS}(\mathcal{K})\leqslant 2^{\lambda}$ .

We prove the theorem through three claims.

Claim 1. If $A\preccurlyeq B$ and $b\in B$ , where $A,B\in\mathbf{K}$ , then there is $C\in\mathbf{K}$ such that

(a)

$C\preccurlyeq B$ and $|C|\leqslant 2^{\lambda}$ ;
(b)

$b\in C$ ;
(c)

$\langle AC\rangle_{B}\preccurlyeq B$ .

Proof of Claim 1. Suppose, toward a contradiction, that no such $C$ exists. In particular, $|B|>2^{\lambda}$ , since otherwise one could take $C=B$ .

We recursively construct an increasing continuous $\preccurlyeq$ -chain $(C_{i})_{i<(2^{\lambda})^{+}}$ in $\mathbf{K}$ , formulas $(\varphi_{i}(\bar{x}_{i}))_{i<(2^{\lambda})^{+}}$ in $\Sigma$ , and tuples $((\bar{a}_{i},\bar{c}_{i}))_{i<(2^{\lambda})^{+}}$ , so that for every $i<(2^{\lambda})^{+}$ :

(1)

$|C_{i}|=2^{\lambda}$ ;
(2)

$b\in C_{0}$ and $C_{i}\preccurlyeq B$ ;
(3)

$\bar{a}_{i}\in A^{|\bar{x}_{i}|}$ and $\bar{c}_{i}\in C_{i}^{|\bar{x}_{i}|}$ ;
(4)

$\langle AC_{i+1}\rangle_{B}\models\varphi_{i}^{*}(\bar{a}_{i},\bar{c}_{i})$ and $\langle AC_{i}\rangle_{B}\not\models\varphi_{i}^{*}(\bar{a}_{i},\bar{c}_{i})$ .

The construction follows from the Löwenheim–Skolem–Tarski axiom and Remark 4.37. At a successor step, we make sure $C_{i+1}$ has the at most $\lambda$ witnesses for the existential quantifiers in $\varphi_{i}^{*}(\bar{x}_{i}^{\prime},\bar{x}_{i}^{\prime\prime})$ . It is in this construction where we use in a key way the assumption that $\mathbf{K}$ is closed under submodules.

Let $X_{1}$ be the set of limit ordinals below $(2^{\lambda})^{+}$ . For each $i\in X_{1}$ , let $\alpha_{i}<i$ be least such that $\bar{c}_{i}\in C_{\alpha_{i}}^{|\bar{x}_{i}|}$ ; such an $\alpha_{i}$ exists because $\bar{c}_{i}$ is finite and the chain is continuous. By Fodor’s lemma, there are a stationary set $X_{2}\subseteq X_{1}$ and an ordinal $\alpha$ such that $\alpha_{i}=\alpha$ for every $i\in X_{2}$ .

Let $\Phi:X_{2}\to\Sigma\times C_{\alpha}^{<\omega}$ given by $\Phi(i)=(\varphi_{i}(\bar{x}_{i}),\bar{c}_{i})$ . The map has range of cardinality at most $2^{\lambda}$ by Remark 4.33. Hence, it follows from the pigeonhole principle that there are a set $X\subseteq X_{2}$ of cardinality $(2^{\lambda})^{+}$ , a formula $\varphi(\bar{x})\in\Sigma$ , and a tuple $\bar{c}\in C_{\alpha}^{|\bar{x}|}$ such that $\varphi_{i}(\bar{x}_{i})=\varphi(\bar{x})$ and $\bar{c}_{i}=\bar{c}$ for every $i\in X$ .

Choose $i<j$ both in $X$ . By Condition (4),

\langle AC_{i+1}\rangle_{B}\models\varphi^{*}(\bar{a}_{i},\bar{c})\quad\text{and}\quad\langle AC_{j+1}\rangle_{B}\models\varphi^{*}(\bar{a}_{j},\bar{c}).

Since $i<j$ , Proposition 2.8(1) gives $\langle AC_{j+1}\rangle_{B}\models\varphi^{*}(\bar{a}_{i},\bar{c}).$ By Proposition 2.8(2), $\langle AC_{j+1}\rangle_{B}\models\varphi^{*}(\bar{a}_{j}-\bar{a}_{i},\bar{c}-\bar{c})=\varphi^{*}(\bar{a}_{j}-\bar{a}_{i},\bar{0}),$ and hence $B\models\varphi^{*}(\bar{a}_{j}-\bar{a}_{i},\bar{0})$ . Since $A\preccurlyeq B$ , Proposition 4.36 yields $A\models\varphi^{*}(\bar{a}_{j}-\bar{a}_{i},\bar{0}).$ Combining this with $\langle AC_{i+1}\rangle_{B}\models\varphi^{*}(\bar{a}_{i},\bar{c})$ and applying Proposition 2.8(1) and (2), we obtain $\langle AC_{i+1}\rangle_{B}\models\varphi^{*}(\bar{a}_{j},\bar{c}).$ Because $i+1<j$ as $j$ is a limit ordinal, Proposition 2.8(1) implies $\langle AC_{j}\rangle_{B}\models\varphi^{*}(\bar{a}_{j},\bar{c}_{j}),$ contradicting Condition (5). This proves Claim 1.

Claim 2. Suppose that $A\preccurlyeq B$ and $A\preccurlyeq B^{\prime}$ , with $b\in B$ and $b^{\prime}\in B^{\prime}$ . Let $C\preccurlyeq B$ and $C^{\prime}\preccurlyeq B^{\prime}$ satisfy Conditions (a)–(c) of Claim 1 for $b\in B$ and $b^{\prime}\in B^{\prime}$ , respectively. If $A\cap C=A\cap C^{\prime}$ and there is an isomorphism $f:C\to C^{\prime}$ such that

f\restriction(A\cap C)=\operatorname{id}_{A\cap C}\quad\text{and}\quad f(b)=b^{\prime},

then $\mathbf{gtp}_{\mathcal{K}}(b/A;B)=\mathbf{gtp}_{\mathcal{K}}(b^{\prime}/A;B^{\prime}).$

Proof of Claim 2. Let $P_{C}:=A\oplus_{A\cap C}C$ be the pusout and $t_{C}:P_{C}\to\langle AC\rangle_{B}$ be the map given by the universal mapping property, i.e., $t_{C}[(a,c)]_{A\cap C}=a+c$ .

The map $t_{C}$ is an epimorphism, we show it is a monomorphism. If $t_{C}([(a,c)]_{A\cap C})=0$ , then $a=-c\in A\cap C$ , and therefore $[(a,c)]_{A\cap C}=0$ . Thus $t_{C}$ is an isomorphism. Its inverse is $g_{C}:\langle AC\rangle_{B}\longrightarrow P_{C}$ given by $g_{C}(a+c)=[(a,c)]_{A\cap C}.$

Similarly, the map $t_{C^{\prime}}:A\oplus_{A\cap C^{\prime}}C^{\prime}\longrightarrow\langle AC^{\prime}\rangle_{B^{\prime}}$ given by $t_{C^{\prime}}([(a,c^{\prime})]_{A\cap C^{\prime}})=a+c^{\prime}$ , is an isomorphism.

Let $h:A\oplus_{A\cap C}C\longrightarrow A\oplus_{A\cap C^{\prime}}C^{\prime}$ given by $h([(a,c)]_{A\cap C})=[(a,f(c))]_{A\cap C^{\prime}}.$ This is well defined because $A\cap C=A\cap C^{\prime}$ and $f$ is the identity on this common submodule. Its inverse is induced by $f^{-1}$ , so $h$ is an isomorphism. Consequently,

s:=t_{C^{\prime}}\circ h\circ g_{C}:\langle AC\rangle_{B}\longrightarrow\langle AC^{\prime}\rangle_{B^{\prime}}

is an isomorphism satisfying $s\restriction A=\operatorname{id}_{A}$ and $s(b)=b^{\prime}$ .

Since $\mathbf{K}$ is closed under submodules, both $\langle AC\rangle_{B},\langle AC^{\prime}\rangle_{B^{\prime}}\in\mathbf{K}$ . So $(b,A,\langle AC\rangle_{B})E^{\mathrm{at}}_{\mathcal{K}}(b^{\prime},A,\langle AC^{\prime}\rangle_{B^{\prime}})$ .

Moreover, Condition (c) of Claim 1 gives $\langle AC\rangle_{B}\preccurlyeq B$ and $\langle AC^{\prime}\rangle_{B^{\prime}}\preccurlyeq B^{\prime}.$ Thus

	$\displaystyle(b,A,B)$	$\displaystyle E^{\mathrm{at}}_{\mathcal{K}}\ (b,A,\langle AC\rangle_{B})$
		$\displaystyle E^{\mathrm{at}}_{\mathcal{K}}\ (b^{\prime},A,\langle AC^{\prime}\rangle_{B^{\prime}})$
		$\displaystyle E^{\mathrm{at}}_{\mathcal{K}}\ (b^{\prime},A,B^{\prime}),$

which proves Claim 2.

Claim 3. If $\theta^{2^{\lambda}}=\theta$ , then $\mathcal{K}$ is $\theta$ -stable.

Proof of Claim 3. Let $A\in\mathbf{K}_{\theta}$ . Suppose, toward a contradiction, that $\{p_{i}=\mathbf{gtp}_{\mathcal{K}}(b_{i}/A;B_{i})\mid i<\theta^{+}\}$ is a family of distinct types in $\mathcal{S}_{\mathcal{K}}(A)$ . For each $i<\theta^{+}$ , choose $C_{i}\preccurlyeq B_{i}$ satisfying Conditions (a)–(c) of Claim 1 for $b_{i}\in B_{i}$ .

Since there are at most $\theta^{2^{\lambda}}=\theta$ subsets of $A$ of cardinality at most $2^{\lambda}$ , by the pigeonhole principle there are a set $S\subseteq\theta^{+}$ of cardinality $\theta^{+}$ and a set $D\subseteq A$ , with $|D|\leqslant 2^{\lambda}$ , such that $A\cap C_{i}=D$ for every $i\in S$ . Fix an enumeration $D=\{d_{\alpha}\mid\alpha<\mu\}$ , where $\mu\leqslant 2^{\lambda}$ .

Expand the language of $R$ -modules by constants $k$ and $(e_{\alpha})_{\alpha<\mu}$ . Let $\Delta$ be the set of isomorphism types of structures $(N,k^{N},(e_{\alpha}^{N})_{\alpha<\mu}),$ where $N\in\mathbf{K}$ , $|N|\leqslant 2^{\lambda}$ , $k^{N}\in N$ , and $e_{\alpha}^{N}\in N$ for every $\alpha<\mu$ . Then $|\Delta|\leqslant 2^{2^{\lambda}}\leqslant\theta^{2^{\lambda}}=\theta.$ For $i\in S$ , assign to $i$ the isomorphism type of $(C_{i},b_{i},(d_{\alpha})_{\alpha<\mu}).$ By the pigeonhole principle, there are distinct $i,j\in S$ and an isomorphism

f:(C_{i},b_{i},(d_{\alpha})_{\alpha<\mu})\cong(C_{j},b_{j},(d_{\alpha})_{\alpha<\mu}).

Thus $f:C_{i}\cong C_{j}$ , $f(b_{i})=b_{j}$ and $f$ fixes $D=A\cap C_{i}=A\cap C_{j}$ pointwise. Claim 2 gives $p_{i}=p_{j}$ , a contradiction. This proves Claim 3.

For every cardinal $\mu\geqslant 2^{\lambda}$ , the cardinal $\theta=2^{\mu}$ satisfies $\theta^{2^{\lambda}}=\theta$ . Hence Claim 3 yields stability in unboundedly many cardinals, and $\mathcal{K}$ is stable. ∎

The following consequence is immediate from Lemma 4.34 and Theorem 4.38.

Corollary 4.39.

We can strengthen Corollary 4.39 as follows.

Corollary 4.40.

Let $\Sigma\subseteq\Lambda^{\infty,\aleph_{0}}_{\infty}$ be a set. If $\leqslant_{\Sigma}\in\mathscr{L}_{R}$ , then $(R\text{-}\mathrm{Mod},\leqslant_{\Sigma})$ is stable.

Proof.

By Proposition 2.5, there is an E-nice set $\Sigma^{\prime}$ such that $\leqslant_{\Sigma}=\leqslant_{\Sigma^{\prime}}$ . The conclusion now follows from Theorem 4.38. ∎

Corollary 4.41.

If $\Sigma$ is a set of first-order $\mathrm{pp}$ -formulas, then $(R\text{-}\mathrm{Mod},\leqslant_{\Sigma})$ is a stable AEC.

Proof.

This follows from Proposition 4.4 and Corollary 4.40. ∎

Thus Corollary 4.41 partially answers Question 4.19 by extending Lemma 4.17.

Remark 4.42.

Theorem 4.38 also yields stability for other interesting AECs of modules. For example, it applies to the class of torsion abelian groups with pure embeddings and to the class of torsion-free abelian groups with pure embeddings, because both underlying classes are closed under submodules. These results were first obtained in [18, Lemma 3.5] and [3, Theorem 0.3] respectively.

5. The lattice over $\mathbb{Z}$

In this section, we study the lattice $\mathscr{L}_{\mathbb{Z}}$ . The first subsection concerns $\mathscr{L}^{1}_{\mathbb{Z}}$ ; the second establishes that $\mathscr{L}_{\mathbb{Z}}$ is a proper class; and the third investigates the failure of amalgamation.

5.1. The sublattice $\mathscr{L}^{1}_{\mathbb{Z}}$

We begin with two elementary order-theoretic observations.

Lemma 5.1.

There is an uncountable antichain in $\mathscr{L}^{1}_{\mathbb{Z}}$ .

Proof.

Let $\mathcal{A}$ be an almost disjoint family of infinite subsets of the set of prime numbers such that $|\mathcal{A}|=2^{\aleph_{0}}$ . Thus every $U\in\mathcal{A}$ is countably infinite, while $U\cap V$ is finite whenever $U,V\in\mathcal{A}$ are distinct. Such a family exists by for example [14, Lemma 9.21].

For $U\in\mathcal{A}$ , let $A\leqslant_{U}B$ if and only if $A\leqslant B$ and for every $p\in U$ , $pB\cap A=pA$ . For every $U\in\mathcal{A}$ , we have that $\leqslant_{U}$ is pleasant with respect to $\Sigma_{U}=\{\exists y(py=x)\,\mid\,p\in U\}$ . Hence $\leqslant_{U}\in\mathscr{L}^{1}_{\mathbb{Z}}$ and $(\mathbb{Z}\text{-}\mathrm{Mod},\leqslant_{U})$ is $(<\aleph_{0})$ -tame and stable.

We show that $\{\leqslant_{U}\mid U\in\mathcal{A}\}$ is an antichain. Let $U,V\in\mathcal{A}$ be distinct. Since $U$ and $V$ are infinite and $U\cap V$ is finite, neither is contained in the other. We prove that $\leqslant_{V}\not\subseteq\leqslant_{U}$ ; the reverse non-inclusion is analogous.

Choose $p\in U\setminus V$ , and let $G=\mathbb{Z}$ and $H=\langle\frac{1}{p}\rangle$ . Clearly $G\not\leqslant_{U}H$ . To see that $G\leqslant_{V}H$ , fix $q\in V$ and $n\in qH\cap G$ . Then there is $a\in\mathbb{Z}$ such that $n=\frac{qa}{p}$ , so $pn=qa$ . Since $p\neq q$ , it follows that $q\mid n$ in $\mathbb{Z}$ . Thus $n\in q\mathbb{Z}=qG$ , and therefore $G\leqslant_{V}H$ . ∎

The following natural question remains open.

Question 5.2.

Is there an antichain of cardinality $2^{2^{\aleph_{0}}}$ in $\mathscr{L}^{1}_{\mathbb{Z}}$ ? This is the largest possible cardinality, since $|\mathscr{L}^{1}_{\mathbb{Z}}|\leqslant 2^{2^{\aleph_{0}}}$ by Proposition 4.3.

Proposition 5.3.

There is a countable strictly increasing chain in $\mathscr{L}^{1}_{\mathbb{Z}}$ .

Proof.

Fix a prime $p$ . For $n<\omega$ , define $A\leqslant_{p,n}B$ if and only if $A\leqslant B$ and $p^{k}B\cap A=p^{k}A$ for every $k\in\{0,\ldots,n\}$ . The relation $\leqslant_{p,n}$ is pleasant with respect to $\Sigma_{p,n}=\{\exists y\,(p^{k}y=x)\mid 0\leqslant k\leqslant n\}$ and hence $\leqslant_{p,n}\in\mathscr{L}^{1}_{\mathbb{Z}}$ . The relations $(\leqslant_{p,n})_{n<\omega}$ form a strictly increasing chain in $\mathscr{L}^{1}_{\mathbb{Z}}$ . ∎

It is likewise open whether $\mathscr{L}^{1}_{\mathbb{Z}}$ contains an uncountable strictly increasing chain.

We next show that not all the strong submodel relations in $\mathscr{L}_{\mathbb{Z}}$ are of the syntactic form so far considered in this paper.

Definition 5.4.

A binary relation $\preccurlyeq$ on $R\text{-}\mathrm{Mod}$ is positive syntactic with respect to $\Sigma$ if $\Sigma\subseteq\Lambda^{\infty,\infty}_{\infty}$ and $\preccurlyeq=\leqslant_{\Sigma}$ . We say $\preccurlyeq$ is positive syntactic if it is positive syntactic with respect to some such $\Sigma$ .

Pleasant and E-nice relations are positive syntactic. Thus every relation introduced so far is positive syntactic. The next theorem shows that this need not hold for an arbitrary element of $\mathscr{L}_{\mathbb{Z}}$ .

Theorem 5.5.

There is a relation $\preccurlyeq\in\mathscr{L}^{1}_{\mathbb{Z}}$ that is not positive syntactic.

Proof.

For $A,B\in\mathbb{Z}\text{-}\mathrm{Mod}$ , define $A\preccurlyeq B$ if and only if $A\leqslant B$ and, for every element $a\in A$ of infinite order and every integer $0<n<\omega$ , $A\models\exists y\,(ny=a)$ if and only if $B\models\exists y\,(ny=a).$ It is straightforward to verify that $(\mathbb{Z}\text{-}\mathrm{Mod},\preccurlyeq)$ is an AEC and that $\preccurlyeq$ lies below $\leqslant_{\mathrm{pp}}$ in $\mathscr{L}_{\mathbb{Z}}$ .

Suppose, toward a contradiction, that $\preccurlyeq=\leqslant_{\Sigma}$ for some $\Sigma\subseteq\Lambda^{\infty,\infty}_{\infty}$ . Fix a prime $p$ , let $B=\mathbb{Z}/p^{2}\mathbb{Z}$ , and let $A=pB$ . Since $A$ has no elements of infinite order, $A\preccurlyeq B$ , and therefore $A\leqslant_{\Sigma}B$ . By applying Proposition 2.8 to the canonical inclusions and projections and adding, we obtain $A\oplus\mathbb{Z}\leqslant_{\Sigma}B\oplus\mathbb{Z}.$ Hence $A\oplus\mathbb{Z}\preccurlyeq B\oplus\mathbb{Z}$ .

The element $(p+p^{2}\mathbb{Z},p)\in A\oplus\mathbb{Z}$ has infinite order. It is divisible by $p$ in $B\oplus\mathbb{Z}$ , but it is not divisible by $p$ in $A\oplus\mathbb{Z}$ . This contradicts that $A\oplus\mathbb{Z}\leqslant_{\Sigma}B\oplus\mathbb{Z}.$ ∎

Remark 5.6.

The relation from Theorem 5.5 does not have the amalgamation property. The span

p(\mathbb{Z}/p^{2}\mathbb{Z})\preccurlyeq\mathbb{Z}/p^{2}\mathbb{Z},\quad p(\mathbb{Z}/p^{2}\mathbb{Z})\preccurlyeq p(\mathbb{Z}/p^{2}\mathbb{Z})\oplus\mathbb{Z}

cannot be completed to a commutative square of strong embeddings. We do not know whether $(\mathbb{Z}\text{-}\mathrm{Mod},\preccurlyeq)$ is stable.

5.2. $\mathscr{L}_{\mathbb{Z}}$ is a proper class

We recall several classical notions from abelian group theory.

Definition 5.7 ([9, p. 299]).

Let $A\in\mathbb{Z}\text{-}\mathrm{Mod}$ and let $p$ be a prime. For every ordinal $\alpha$ , define $p^{\alpha}A$ by induction as follows:

(1)

$p^{0}A=A$ ;
(2)

$p^{\beta+1}A=p(p^{\beta}A)$ ;
(3)

if $\alpha$ is a limit ordinal, then $p^{\alpha}A=\bigcap_{\beta<\alpha}p^{\beta}A$ .

Definition 5.8 ([9, p. 386]).

Let $\alpha$ be an ordinal and let $p$ be a prime. For $A,B\in\mathbb{Z}\text{-}\mathrm{Mod}$ , we say that $A$ is $p^{\alpha}$ -isotype in $B$ , denoted by $A\leqslant^{p^{\alpha}}_{\mathrm{iso}}B$ , if $A\leqslant B$ and $p^{\beta}B\cap A=p^{\beta}A$ for every $\beta\leqslant\alpha$ .

Proposition 5.9.

Let $\alpha$ be an ordinal and let $p$ be a prime. The relation $\leqslant^{p^{\alpha}}_{\mathrm{iso}}$ is E-nice with respect to a set

\{\psi_{\beta}(x_{0})\mid\beta\leqslant\alpha\}\subseteq\Lambda^{|\alpha|^{+}+\aleph_{0},\,|\alpha|^{+}+\aleph_{0}}_{1}.

Moreover, $\leqslant^{(|\alpha|^{+}+\aleph_{0},\infty)}_{\mathrm{pp}}\subseteq\leqslant^{p^{\alpha}}_{\mathrm{iso}}$ , and $\leqslant^{p^{\alpha}}_{\mathrm{iso}}$ refines the direct summand relation.

Proof.

We recursively define formulas $\varphi_{\beta}(x_{0})$ for $\beta\leqslant\alpha$ .

Let $\varphi_{0}(x_{0}):=x_{0}=x_{0}$ .

If $\beta=\gamma+1$ , let

\varphi_{\gamma+1}(x_{0}):=\exists v\,(pv=x_{0}\wedge\varphi_{\gamma}(v)).

If $\beta$ is a limit ordinal, let

\varphi_{\beta}(x_{0}):=\bigwedge_{\gamma<\beta}\varphi_{\gamma}(x_{0}).

Put $\Delta=\{\varphi_{\beta}(x_{0})\mid\beta\leqslant\alpha\}$ . Then $A\leqslant^{p^{\alpha}}_{\mathrm{iso}}B$ if and only if $A\leqslant_{\Delta}B.$

For every $\beta\leqslant\alpha$ , $\varphi_{\beta}(x_{0})\in\Lambda^{|\beta|^{+}+\aleph_{0},\aleph_{0}}_{\beta}$ . Proposition 2.5 therefore gives an E-nice formula $\psi_{\beta}(x_{0})\in\Lambda^{|\beta|^{+}+\aleph_{0},\,|\beta|^{+}+\aleph_{0}}_{1}$ equivalent to $\varphi_{\beta}(x_{0})$ for every $\beta\leqslant\alpha$ . Hence $\leqslant^{p^{\alpha}}_{\mathrm{iso}}=\leqslant_{\{\psi_{\beta}(x_{0})\mid\beta\leqslant\alpha\}}$ , so $\leqslant^{p^{\alpha}}_{\mathrm{iso}}$ is E-nice.

The moreover part follows because each $\varphi_{\beta}(x_{0})\in\Lambda^{|\alpha|^{+}+\aleph_{0},\aleph_{0}}_{\infty}$ . The final assertion follows from Corollary 2.10. ∎

Lemma 5.10.

Let $\alpha$ be an ordinal and let $p$ be a prime. Then $(\mathbb{Z}\text{-}\mathrm{Mod},\leqslant^{p^{\alpha}}_{\mathrm{iso}})$ is an AEC with

\mathrm{LS}(\mathbb{Z}\text{-}\mathrm{Mod},\leqslant^{p^{\alpha}}_{\mathrm{iso}})\leqslant|\alpha|+\aleph_{0}.

Proof.

It is clear that $(\mathbb{Z}\text{-}\mathrm{Mod},\leqslant^{p^{\alpha}}_{\mathrm{iso}})$ is an abstract class. Transitivity follows from Proposition 5.9, or directly from [9, p. 365, Condition (b)]. We verify Conditions (4)–(6) of Definition 2.14.

(4)

Let $(A_{i})_{i<\delta}$ be an increasing continuous $\leqslant^{p^{\alpha}}_{\mathrm{iso}}$ -chain, and let $A:=\bigcup_{i<\delta}A_{i}$ . Condition (4.1) is clear, and Condition (4.2) is [9, p. 365, Condition (c)]. To verify Condition (4.3), suppose that $A_{i}\leqslant^{p^{\alpha}}_{\mathrm{iso}}B$ for every $i<\delta$ . Fix $\beta\leqslant\alpha$ . The inclusion $p^{\beta}A\subseteq p^{\beta}B\cap A$ is clear. Conversely, if $a\in p^{\beta}B\cap A$ , choose $i<\delta$ with $a\in A_{i}$ . Then $a\in p^{\beta}B\cap A_{i}=p^{\beta}A_{i}\subseteq p^{\beta}A.$ Thus $A\leqslant^{p^{\alpha}}_{\mathrm{iso}}B$ .
(5)

Coherence follows from Proposition 5.9, or directly from [9, p. 365, Condition (a)].
(6)

Let $X\subseteq B\in\mathbb{Z}\text{-}\mathrm{Mod}$ , and let $\kappa:=|\alpha|+\aleph_{0}$ . Let $\Sigma=\{\psi_{\beta}(x_{0})\mid\beta\leqslant\alpha\}$ be the E-nice set given by Proposition 5.9. We construct an increasing chain $(A_{n})_{n<\omega}$ of submodules of $B$ . Set $A_{0}=\langle X\rangle$ . Having defined $A_{n}$ , for every $a\in A_{n}$ and $\beta\leqslant\alpha$ such that $B\models\psi_{\beta}(a)$ , choose in $B$ a tuple of witnesses for the existential quantifiers of $\psi_{\beta}(a)$ , and let $A_{n+1}$ be the submodule generated by $A_{n}$ together with all these witnesses. At each stage, at most $|A_{n}|+\kappa$ tuples, each of length at most $\kappa$ , are added. Hence $|A_{n}|\leqslant|X|+\kappa$ for every $n<\omega$ .

Let $A:=\bigcup_{n<\omega}A_{n}$ . Then $X\subseteq A\leqslant B$ and $|A|\leqslant|X|+\kappa$ . If $a\in A$ and $B\models\psi_{\beta}(a)$ , the required witnesses were added at a later stage, so $A\models\psi_{\beta}(a)$ . The converse follows from Proposition 2.8. Thus $A\leqslant_{\Sigma}B$ , and Proposition 5.9 gives $A\leqslant^{p^{\alpha}}_{\mathrm{iso}}B$ .

∎

Corollary 5.11.

For every ordinal $\alpha$ and every prime $p$ , the AEC $(\mathbb{Z}\text{-}\mathrm{Mod},\leqslant^{p^{\alpha}}_{\mathrm{iso}})$ is stable.

Proof.

This follows from Proposition 5.9, Lemma 5.10, and Theorem 4.38. ∎

Recall that the $p$ -length of a $\mathbb{Z}$ -module $A$ , denoted by $\operatorname{len}_{p}(A)$ , is the least ordinal $\alpha$ such that $p^{\alpha+1}A=p^{\alpha}A$ .

Proposition 5.12.

Let $p$ be a prime. If $\theta$ is an infinite cardinal, then $\mathrm{LS}(\mathbb{Z}\text{-}\mathrm{Mod},\leqslant^{p^{\theta}}_{\mathrm{iso}})=\theta$ .

Proof.

Lemma 5.10 gives $\mathrm{LS}(\mathbb{Z}\text{-}\mathrm{Mod},\leqslant^{p^{\theta}}_{\mathrm{iso}})\leqslant\theta$ . Suppose, toward a contradiction, that $\mathrm{LS}(\mathbb{Z}\text{-}\mathrm{Mod},\leqslant^{p^{\theta}}_{\mathrm{iso}})=\lambda<\theta$ .

By [9, Theorem 10.1.6], there is a $p$ -group $B$ of $p$ -length $\theta+1$ . Choose $b\in p^{\theta}B\setminus p^{\theta+1}B$ . The Löwenheim–Skolem–Tarski axiom yields $A\leqslant^{p^{\theta}}_{\mathrm{iso}}B$ such that $b\in A$ and $|A|\leqslant\lambda<\theta$ . By cardinality considerations, $\operatorname{len}_{p}(A)<|A|^{+}\leqslant\theta$ . On the other hand, $\operatorname{len}_{p}(A)\geqslant\theta+1$ . Indeed, if $p^{\alpha}A=p^{\alpha+1}A$ for some $\alpha<\theta+1$ , then an easy induction gives that $p^{\theta}A=p^{\theta+1}A$ . This is impossible, since $b\in p^{\theta}B\cap A=p^{\theta}A$ but $b\notin p^{\theta+1}A$ as $b\notin p^{\theta+1}B$ . This contradiction proves the result. ∎

Theorem 5.13.

The lattice $\mathscr{L}_{\mathbb{Z}}$ is a proper class. More precisely, for every fixed prime $p$ , the relations $\{\leqslant^{p^{\theta}}_{\mathrm{iso}}\mid\theta\in\mathrm{Card}\}$ form a strictly increasing proper-class-sized chain in $\mathscr{L}_{\mathbb{Z}}$ .

Proof.

The relations become stronger as $\theta$ increases, and Proposition 5.12 shows that relations indexed by distinct infinite cardinals have distinct Löwenheim–Skolem numbers. Hence the chain is strict and proper-class-sized. ∎

Remark 5.14.

Let $\alpha$ be an ordinal and let $p$ be a prime. Define $A\leqslant^{p^{\alpha}}_{\mathrm{iso},\mathrm{pp}}B$ if and only if $A\leqslant^{p^{\alpha}}_{\mathrm{iso}}B$ and $A\leqslant_{\mathrm{pp}}B$ .

All results proved above for $\leqslant^{p^{\alpha}}_{\mathrm{iso}}$ extend to $\leqslant^{p^{\alpha}}_{\mathrm{iso},\mathrm{pp}}$ . In particular, $\{\leqslant^{p^{\theta}}_{\mathrm{iso},\mathrm{pp}}\mid\theta\in\mathrm{Card}\}$ forms a strictly increasing proper-class-sized chain in $\mathscr{L}^{2}_{\mathbb{Z}}$ and $\mathscr{L}^{2}_{\mathbb{Z}}$ is a proper class.

5.3. Failure of amalgamation

We conclude by showing that amalgamation often fails in $\mathscr{L}_{\mathbb{Z}}$ . More precisely, if $\preccurlyeq\in\mathscr{L}_{\mathbb{Z}}$ is at least as strong as $\leqslant^{\mathrm{weak}}_{\mathrm{pp,\infty}}$ and satisfies a suitable syntactic hypothesis, then $(\mathbb{Z}\text{-}\mathrm{Mod},\preccurlyeq)$ fails amalgamation. This indicates that the model theory of strong submodel relations in $\mathscr{L}^{2}_{\mathbb{Z}}$ is more complicated than that of $(\mathbb{Z}\text{-}\mathrm{Mod},\leqslant_{\mathrm{pp}})$ .

Definition 5.15.

The abstract class $(R\text{-}\mathrm{Mod},\preccurlyeq)$ is closed under pushouts if, for every $A\preccurlyeq B,C$ , the maps of the pushout $(B\oplus_{A}C,f:B\to B\oplus_{A}C,g:C\to B\oplus_{A}C)$ are strong embeddings.

Proposition 5.16.

Let $\preccurlyeq\in\mathscr{L}_{R}$ be positive syntactic. $(R\text{-}\mathrm{Mod},\preccurlyeq)$ has the amalgamation property if and only if $(R\text{-}\mathrm{Mod},\preccurlyeq)$ is closed under pushouts.

Proof.

If $(R\text{-}\mathrm{Mod},\preccurlyeq)$ is closed under pushouts, then it clearly has amalgamation. Conversely, assume amalgamation, and let $A\preccurlyeq B,C$ . Choose an amalgam $D$ and strong embeddings $h_{B}:B\to D$ and $h_{C}:C\to D$ satisfying $h_{B}\restriction A=h_{C}\restriction A.$ We show that the canonical map $f:B\to B\oplus_{A}C$ is a strong embedding; the argument for $g:C\to B\oplus_{A}C$ is analogous.

Choose $\Sigma\subseteq\Lambda^{\infty,\infty}_{\infty}$ such that $\preccurlyeq=\leqslant_{\Sigma}$ . By Remark 2.11, $f$ is a monomorphism. It remains to show that, for every $\varphi(\bar{x})\in\Sigma$ and every $\bar{b}\in B^{|\bar{x}|}$ , $B\oplus_{A}C\models\varphi(f(\bar{b}))$ if and only if $B\models\varphi(\bar{b}).$

The reverse implication follows from Proposition 2.8. For the forward implication, suppose that $B\oplus_{A}C\models\varphi(f(\bar{b}))$ . By the universal mapping property of the pushout, there is a homomorphism $k:B\oplus_{A}C\to D$ such that $k\circ f=h_{B}$ and $k\circ g=h_{C}$ . Proposition 2.8 gives $D\models\varphi(h_{B}(\bar{b})).$ Since $h_{B}$ is a strong embedding, $B\models\varphi(\bar{b})$ . Thus $f$ is a strong embedding. ∎

Theorem 5.17.

If $\preccurlyeq\in\mathscr{L}_{\mathbb{Z}}$ is positive syntactic and lies above $\leqslant^{\mathrm{weak}}_{\mathrm{pp,\infty}}$ in $\mathscr{L}_{\mathbb{Z}}$ , then $(\mathbb{Z}\text{-}\mathrm{Mod},\preccurlyeq)$ does not have the amalgamation property.

Proof.

Let $\preccurlyeq\in\mathscr{L}_{\mathbb{Z}}$ be positive syntactic and lie above $\leqslant^{\mathrm{weak}}_{\mathrm{pp,\infty}}$ . We show that $(\mathbb{Z}\text{-}\mathrm{Mod},\preccurlyeq)$ is not closed under pushouts; Proposition 5.16 then gives the result.

Choose prime numbers $(p_{i})_{i<\omega}$ such that

p_{i+1}>p_{i}^{2}(\prod_{j=0}^{i-1}p_{j})

for every $i<\omega$ . For each $f:\omega\to\mathbb{Z}$ , let $\hat{f}:\omega\to\mathbb{Z}/p_{i}\mathbb{Z}$ be given by $\hat{f}(i)=f(i)+p_{i}\mathbb{Z}$ .

Define

(i)

$M=\prod_{i<\omega}\mathbb{Z}/p_{i}\mathbb{Z}$ ;
(ii)

$A=\{\hat{f}\mid f:\omega\to\mathbb{Z}\text{ and }(\exists k<\omega)(\forall i>k)\ f(i)=0\};$
(iii)

$B=\{\hat{f}\mid f:\omega\to\mathbb{Z}\text{ and }(\exists k<\omega)(\forall i>k)\ f(i+1)=p_{i}f(i)\};$

(iv)

\begin{split}C=\{\hat{f}\mid f:\omega\to\mathbb{Z}\text{ and }{}&(\exists k<\omega)(\forall i>k)\\ &\begin{cases}f(i)=0&\text{if $i$ is odd},\\ f(i+2)=p_{i}p_{i+1}f(i)\text{ in }\mathbb{Z}&\text{if $i$ is even}\end{cases}\}.\end{split}

Notice that $A\leqslant B,C\leqslant M$ . For $n<\omega$ , let

A(n)=\{\hat{f}\mid f:\omega\to\mathbb{Z}\text{ and }(\forall i>n)\ f(i)=0\}.

Then:

(1)

$A(n)\leqslant_{\oplus}B$ and $A(n)\leqslant_{\oplus}C$ for every $n<\omega$ . Indeed, the intended complements are

$D(n)=\{\hat{f}\mid f:\omega\to\mathbb{Z}\text{ , }\hat{f}\in B\text{ and }(\forall i\leqslant n)\ f(i)=0\}$

and

$E(n)=\{\hat{f}\mid f:\omega\to\mathbb{Z}\text{ , }\hat{f}\in C\text{ and }(\forall i\leqslant n)\ f(i)=0\}.$

Thus $A(n)\preccurlyeq B,C$ .
(2)

If $n\leqslant m<\omega$ , then $A(n)\leqslant_{\oplus}A(m)$ , with complement

$A^{\prime}(n)=\{\hat{f}\mid f:\omega\to\mathbb{Z}\text{ , }\hat{f}\in A(m)\text{ and }(\forall i\leqslant n)\ f(i)=0\}.$

Thus $A(n)\preccurlyeq A(m)$ .

Since $A=\bigcup_{n<\omega}A(n)$ , smoothness gives $A\preccurlyeq B,C$ .

Let $f:B\to B\oplus_{A}C$ and $g:C\to B\oplus_{A}C$ be the canonical maps of the pushout (see Remark 2.11). We show that $f$ is not a $\leqslant^{\mathrm{weak}}_{\mathrm{pp,\infty}}$ -embedding and hence is not a $\preccurlyeq$ -embedding.

Let $\varphi(x):=\exists z\,\psi(x,z)$ , where

\psi(x,z):=\bigwedge_{n<\omega}\exists y_{0}\cdots\exists y_{2n-1}\begin{pmatrix}\displaystyle\bigwedge_{j<2n}p_{j}y_{j}=0,\\[2.0pt] \displaystyle\prod_{j<2n}p_{j}\mid x-\sum_{j<2n}y_{j},\\[2.0pt] \displaystyle\prod_{j<2n}p_{j}\mid z-\sum_{\begin{subarray}{c}j<2n\\ j\text{ even}\end{subarray}}y_{j}\end{pmatrix}.

Observe that $\varphi(x)\in\Lambda^{\mathrm{wpp}}_{\infty}$ .

Let $b:\omega\to\mathbb{Z}$ be defined by $b(0)=1$ and $b(i+1)=p_{i}b(i)$ for every $i<\omega$ . Note that $\hat{b}\in B$ and we claim that

B\oplus_{A}C\models\varphi(f(\hat{b}))\quad\text{but}\quad B\not\models\varphi(\hat{b}).

Let $c:\omega\to\mathbb{Z}$ be defined by

c(i)=\begin{cases}0&\text{if $i$ is odd},\\ 1&\text{if $i=0$},\\ p_{i-2}p_{i-1}c(i-2)&\text{if $i\geqslant 2$ is even}.\end{cases}

Note that $\hat{c}\in C$ .

Fix $n<\omega$ . For every $j<2n$ , let $a_{j}:\omega\to\mathbb{Z}$ be given by $a_{j}(j)=b(j)$ and $a_{j}(i)=0$ if $i\neq j$ . Note that $\widehat{a_{j}}\in A$ for every $j<2n$ .

We have that $B\models\bigwedge_{j<2n}p_{j}\widehat{a_{j}}=0$ . Let $b^{\prime}:\omega\to\mathbb{Z}$ be given by

b^{\prime}(i)=\begin{cases}0&\text{if $i<2n$},\\[2.0pt] \displaystyle\frac{b(i)}{p_{0}\cdots p_{2n-1}}&\text{if $i\geqslant 2n$}.\end{cases}

Note that $\hat{b^{\prime}}\in B$ and

B\models\prod_{j<2n}p_{j}\hat{b^{\prime}}=\hat{b}-\sum_{j<2n}\widehat{a_{j}}.

Similarly,

C\models\prod_{j<2n}p_{j}\mid\hat{c}-\sum_{\begin{subarray}{c}j<2n\\ j\text{ even}\end{subarray}}\widehat{a_{j}}.

Using Proposition 2.8 and the identity $f\restriction A=g\restriction A$ , we conclude that $B\oplus_{A}C\models\psi(f(\hat{b}),g(\hat{c})),$ and hence $B\oplus_{A}C\models\varphi(f(\hat{b}))$ .

It remains to show that $B\not\models\varphi(\hat{b})$ . Suppose otherwise, let $e:\omega\to\mathbb{Z}$ such that $\hat{e}\in B$ witnesses the formula, i.e., $B\models\psi(\hat{b},\hat{e})$ . Then there is $k<\omega$ such that $e(i+1)=p_{i}e(i)$ for every $i>k$ . Let $k^{\prime}<\omega$ with $k^{\prime}>k$ such that $|e(k)|<p_{k^{\prime}}$ . Then using that $p_{i+1}>p_{i}^{2}(\prod_{j=0}^{i-1}p_{j})$ for every $i<\omega$ , it follows that for every $i>k^{\prime}$ we have both:

(1)

e(i+1)=p_{i}e(i)\text{ and }|e(i)|<p_{i}

Choose $n$ large enough such that there is an odd integer $m$ with

k^{\prime}<m\quad\text{and}\quad m+1<2n.

Let $(\widehat{a_{j}})_{j<2n}$ be witnesses to the $n$ th conjunct of $\psi(\hat{b},\hat{e})$ .

For every $j<2n$ , it follows from $B\models p_{j}\widehat{a_{j}}=0$ and the fact that the primes $p_{i}$ ’s are distinct, that $p_{i}\mid a_{j}(i)$ for every $i\neq j$ . Hence $\widehat{a_{j}}(i)=0_{\mathbb{Z}/p_{i}\mathbb{Z}}$ for every $i\neq j$ .

For $j<2n$ , the first divisibility condition then yields

(2)

p_{j}\mid b(j)-a_{j}(j).

The second divisibility condition then yields

(3)

\begin{cases}p_{j}\mid e(j)&\text{if $j<2n$ is odd},\\ p_{j}\mid e(j)-a_{j}(j)&\text{if $j<2n$ is even}.\end{cases}

In particular, for $m<2n$ , which is odd, we have that $p_{m}\mid e(m)$ by Equation (3). Since $m>k^{\prime}$ we also have that $|e(m)|<p_{m}$ by Equation (1). Hence $e(m)=0$ . As $e(m+1)=p_{m}e(m)$ by Equation (1), we conclude that $e(m+1)=0$ .

Since $m+1<2n$ is even, it follows that

p_{m+1}\mid e(m+1)-a_{m+1}(m+1)=-a_{m+1}(m+1)

from Equation (3) and the last equation of the previous paragraph. Then it follows that $p_{m+1}\mid b(m+1)$ by Equation (2). This is a contradiction as an easy induction yields that $0<b(m+1)<p_{m+1}$ . ∎

Remark 5.18.

Observe that $f:B\to B\oplus_{A}C$ of the proof of Theorem 5.17 is an explicit example of a pure embedding which is not a $\leqslant^{\mathrm{weak}}_{\mathrm{pp,\infty}}$ -embedding. See Remark 4.28 for another example.

Appendix A Depiction of $\mathscr{L}_{\mathbb{Z}}$

The following figure depicts our current understanding of $\mathscr{L}_{\mathbb{Z}}$ :

Figure 1. Strong submodel relations on

\mathbb{Z}\text{-}\mathrm{Mod}

References

[1] J. T. Baldwin. Categoricity. University Lecture Series, American Mathematical Society, Providence, RI, 2009.
[2] J. T. Baldwin and W. Boney. Hanf numbers and presentation theorems in AECs. In J. Iovino, editor, Beyond First Order Model Theory, CRC Press, 2017, 327–352.
[3] J. T. Baldwin, P. C. Eklof, and J. Trlifaj. ${}^{\perp}N$ as an abstract elementary class. Ann. Pure Appl. Logic 149 (2007), no. 1, 25–39.
[4] W. Boney and S. Vasey. A survey of tame abstract elementary classes. In J. Iovino, editor, Beyond First-Order Model Theory, CRC Press, 2017, 353–427.
[5] W. Boney. A module-theoretic introduction to abstract elementary classes. In A. Martsinkowsky, editor, Functor and Tensor Categories, Models, and Systems, Springer Proceedings in Mathematics & Statistics 521 (2026).
[6] R. Carnevale and G. Paolini. $\mu$ -abstract elementary classes of modules. Preprint, arXiv:2607.12160.
[7] S. Cox. Kaplansky classes and stability. Preprint, arXiv:2608.00905.
[8] J. Feigert, D. Herden and M. Mazari-Armida. On the abstract elementary class of acts with pure embeddings. Preprint, arXiv:2608.12631.
[9] L. Fuchs. Abelian Groups. Springer, Cham, 2015.
[10] R. Göbel and J. Trlifaj. Approximations and Endomorphism Algebras of Modules. 2nd ed., vol. 1, Approximations; vol. 2, Predictions, Walter de Gruyter, Berlin–Boston, 2012.
[11] R. Grossberg. Classification theory for abstract elementary classes. In Y. Zhang, editor, Logic and Algebra, Contemp. Math., vol. 302, American Mathematical Society, 2002, 165–204.
[12] R. Grossberg and M. VanDieren. Galois-stability for tame abstract elementary classes. J. Math. Log. 6 (2006), 25–49.
[13] D. Herden, M. Mazari-Armida, and M. D. Walton. An unstable abstract elementary class of modules: a variation of Paolini–Shelah’s example. Preprint, arXiv:2604.03080, 2026.
[14] T. Jech. Set Theory. 3rd ed., Springer Monographs in Mathematics, Springer, Berlin, 2003.
[15] T. G. Kucera and M. Mazari-Armida. On universal modules with pure embeddings. Math. Log. Q. 66 (2020), 395–408.
[16] D. Marker. Lectures on Infinitary Model Theory. Cambridge University Press, Cambridge, 2016.
[17] M. Mazari-Armida. Superstability, noetherian rings and pure-semisimple rings. Ann. Pure Appl. Logic 172 (2021), no. 3, 102917.
[18] M. Mazari-Armida. A model theoretic solution to a problem of László Fuchs. J. Algebra 567 (2021), 196–209.
[19] M. Mazari-Armida and J. Rosický. Relative injective modules, superstability and noetherian categories. J. Math. Log. 26 (2026), no. 1, 245002.
[20] M. Mazari-Armida and J. Trlifaj. Deconstructible classes of modules and stability. Preprint, arXiv:2512.17799.
[21] G. Paolini and S. Shelah. On the problem of stability of abstract elementary classes of modules. Preprint, arXiv:2512.02545.
[22] M. Prest. Model Theory and Modules. London Mathematical Society Lecture Note Series, vol. 130, Cambridge University Press, Cambridge, 1988.
[23] M. Prest. Purity, Spectra and Localisation. Encyclopedia of Mathematics and its Applications, vol. 121, Cambridge University Press, Cambridge, 2009.
[24] J. Šaroch and J. Trlifaj. Deconstructible abstract elementary classes of modules and categoricity. Bull. Lond. Math. Soc. 56 (2024), 3854–3866.
[25] S. Shelah. Stable theories. Israel J. Math. 7 (1969), 187–202.
[26] S. Shelah. Classification of nonelementary classes II: Abstract elementary classes. In J. T. Baldwin, editor, Classification Theory, Lecture Notes in Mathematics, vol. 1292, Springer, Berlin, 1987, 419–497.
[27] S. Shelah. Categoricity for abstract classes with amalgamation. Ann. Pure Appl. Logic 98 (1999), no. 1, 261–294.
[28] S. Shelah. Modules and infinitary logic. In Groups and Model Theory, Contemp. Math., vol. 576, American Mathematical Society, Providence, RI, 2012, 305–316.
[29] S. Shelah. Universal structures. Notre Dame J. Form. Log. 58 (2017), no. 2, 159–177.
[30] S. Shelah. Classification Theory for Abstract Elementary Classes. Vols. 1 and 2, Mathematical Logic and Foundations, vols. 18 and 20, College Publications, 2009.
[31] M. Ziegler. Model theory of modules. Ann. Pure Appl. Logic 26 (1984), 149–213.

Abstract.

Key words and phrases:

2020 Mathematics Subject Classification

1. Introduction

Theorem 1.1.

Theorem 1.2.

Theorem 1.3.

2. Preliminaries and basic results

2.1. Model theory of modules

Definition 2.1.

Remark 2.2.

Definition 2.3.

Definition 2.4.

Proposition 2.5.

Proof.

Proposition 2.6.

Proof.

Remark 2.7.

Proposition 2.8.

Proof.

Definition 2.9.

Corollary 2.10.

Proof.

Remark 2.11.

2.2. Abstract elementary classes

Definition 2.12.

Definition 2.13.

Definition 2.14.

Remark 2.15.

Definition 2.16.

Definition 2.17.

Definition 2.18.

Definition 2.19.

3. The lattice structure of ℒR\mathscr{L}_{R}

Definition 3.1.

3.1. Basic results

Proposition 3.2.

Proof.

Proposition 3.3.

Proof.

Remark 3.4.

Corollary 3.5.

Proof.

Lemma 3.6.

Proof.

Proposition 3.7.

Proof.

3.2. When is ⩽⊕∈ℒR\leqslant_{\oplus}\in\mathscr{L}_{R}?

Proposition 3.8.

Lemma 3.9.

Proof.

Fact 3.10 ([23, Theorem 4.5.7]).

Theorem 3.11.

Proof.

Question 3.12.

4. The sublattices ℒR1\mathscr{L}^{1}_{R} and ℒR2\mathscr{L}^{2}_{R}

4.1. The sublattice ℒR1\mathscr{L}^{1}_{R}

Definition 4.1.

Remark 4.2.

Proposition 4.3.

Proof.

Proposition 4.4.

Proof.

Example 4.5.

Definition 4.6.

Example 4.7.

Remark 4.8.

Lemma 4.9.

Proof.

Definition 4.10.

Remark 4.11.

Theorem 4.12.

Proof.

Remark 4.13.

Remark 4.14.

Corollary 4.15.

Proposition 4.16.

Proof.

Lemma 4.17.

Proof.

3. The lattice structure of $\mathscr{L}_{R}$

3.2. When is $\leqslant_{\oplus}\in\mathscr{L}_{R}$ ?

4. The sublattices $\mathscr{L}^{1}_{R}$ and $\mathscr{L}^{2}_{R}$

4.1. The sublattice $\mathscr{L}^{1}_{R}$

4.2. The sublattice $\mathscr{L}^{2}_{R}$

5. The lattice over $\mathbb{Z}$

5.1. The sublattice $\mathscr{L}^{1}_{\mathbb{Z}}$

5.2. $\mathscr{L}_{\mathbb{Z}}$ is a proper class