The spectrum of limit models in the abstract elementary class $(\operatorname{Mod}(T),\leq_{\mathrm{pp}})$ of modules with pure embeddings

For a ring $R$ and a first-order theory $T$ extending the theory of $R$-modules such that $\mathbf{K}^T=(\operatorname{Mod}(T),\leq_{\mathrm{pp}})$ has joint embedding and amalgamation, Kucera and Mazari-Armida asked how many non-isomorphic $(\lambda,\alpha)$-limit models of cardinality $\lambda$ the class $\mathbf{K}^T$ can have — its limit-model spectrum — and whether some choice of $R$ and $T$ realizes the a priori maximum of one plus the number of infinite regular cardinals at most $|T|$. The countable-ring case is settled in the source paper itself (eventually a unique limit model under Galois-superstability, otherwise exactly two). Later work settles adjacent settings: the class of all modules with ordinary embeddings (spectrum controlled by parametrized noetherianity, every infinite number realizable), general abstract elementary classes admitting a well-behaved independence relation (long limit models isomorphic, short ones pairwise non-isomorphic), and complete first-order theories (exactly $|\alpha|+1$ limit models when the local character cardinal $\kappa_r(T)=\aleph_\alpha$). The spectrum for pure embeddings at the question's stated generality — uncountable rings, incomplete theories, chain cofinalities strictly between $\omega$ and $|T|^+$ — remains open, in particular whether more than two non-isomorphic limit models can ever occur in such a class and whether the a priori maximum is realizable.

Background

An embedding $f:M\to N$ of left $R$-modules is pure if every finite system of linear equations with parameters from $M$ that is solvable in $N$ is already solvable in $M$; equivalently, $f$ preserves the truth of positive-primitive (pp) formulas, which are existentially quantified conjunctions of linear equations. Purity is the backbone of the model theory of modules: by pp-quantifier elimination, every formula is equivalent, modulo the theory of $R$-modules, to a Boolean combination of pp-formulas and sentences about the cardinal invariants $\operatorname{Inv}(-,\varphi,\psi)$ that count independent solutions of one pp-formula lying outside another.

An abstract elementary class (AEC) is a pair $\mathbf{K}=(K,\leq_{\mathbf{K}})$ consisting of a class of structures in a fixed language and a partial order on it extending the substructure relation, closed under directed colimits, satisfying coherence (if $M_0\leq_{\mathbf{K}}M_2$ and $M_0\subseteq M_1\subseteq M_2$ with $M_1$ in the class, then $M_0\leq_{\mathbf{K}}M_1\leq_{\mathbf{K}}M_2$), and equipped with a downward Löwenheim–Skolem number $\operatorname{LS}(\mathbf{K})$; AECs were introduced in the 1980s as a semantic generalization of elementary classes capable of capturing non-first-order axiomatizable classes. For a ring $R$ and a first-order theory $T$ (not necessarily complete) with an infinite model extending the theory of $R$-modules, the pair $\mathbf{K}^T=(\operatorname{Mod}(T),\leq_{\mathrm{pp}})$, ordered by the pure-submodule relation, is an AEC with $\operatorname{LS}(\mathbf{K}^T)=|T|=|R|+\aleph_0$. Natural examples include the torsion-free abelian groups, the reduced torsion-free abelian groups, and the torsion abelian groups, each with the pure-subgroup relation. The source paper works under the standing hypothesis that $\mathbf{K}^T$ has the joint embedding and amalgamation properties (its Hypothesis 3.9).

In an AEC, Galois types play the role of first-order types, and $\mathbf{K}$ is $\lambda$-Galois-stable if every model of cardinality $\lambda$ has at most $\lambda$ Galois types over it; $\mathbf{K}$ is Galois-superstable if it is $\lambda$-Galois-stable for every sufficiently large $\lambda$. A model $M$ of cardinality $\lambda$ is a $(\lambda,\alpha)$-limit model, for a limit ordinal $\alpha<\lambda^+$, if $M$ is the union of an increasing continuous chain $\langle M_i:i<\alpha\rangle$ of models of cardinality $\lambda$ in which each successor $M_{i+1}$ is universal over its predecessor $M_i$ (every embedding of $M_i$ into a member of the class of cardinality $\lambda$ extends, fixing $M_i$, to an embedding into $M_{i+1}$). Limit models, introduced in the 1990s as substitutes for saturated models in non-elementary classes, exist precisely in the Galois-stability cardinals (given joint embedding, amalgamation and no maximal models), and uniqueness of limit models — all $(\lambda,\alpha)$-limit models isomorphic regardless of $\alpha$ — is equivalent to superstability in tame AECs (tameness: Galois types over a model are determined by their restrictions to submodels of some fixed small cardinality). The number of isomorphism types among the limit models of cardinality $\lambda$, the limit-model spectrum, therefore measures how far the class sits from superstable behavior; the question posed here is the module-theoretic instance of a spectrum problem also pursued abstractly (Conjecture 2 of Boney–VanDieren, and the general-AEC and first-order theorems of Beard and of Beard–Mazari-Armida).

Two facts anchor what is known for $\mathbf{K}^T$. First, a back-and-forth argument shows that $(\lambda,\alpha)$- and $(\lambda,\beta)$-limit models are isomorphic whenever $\operatorname{cf}(\alpha)=\operatorname{cf}(\beta)$, so only the cofinality of the witnessing chain matters. Second, the source paper proves that if $\lambda\ge|T|^+$ and $\operatorname{cf}(\alpha)\ge|T|^+$, then the $(\lambda,\alpha)$-limit model is pure-injective (equivalently algebraically compact: every system of linear equations over it that is finitely solvable is solvable), and all such long limit models are isomorphic; and that, under closure of $\mathbf{K}^T$ under direct sums, a $(\lambda,\omega)$-limit model is isomorphic to $N^{(\aleph_0)}$ for a $(\lambda,|T|^+)$-limit model $N$. Consequently the number of non-isomorphic limit models of cardinality $\lambda$ is at most one plus the number of infinite regular cardinals at most $|T|$ (Remark 4.11 of the source) — at most two when $R$ is countable. For countable rings the source paper determines the spectrum completely: under Galois-superstability there is a unique limit model of each sufficiently large cardinality, and otherwise there are exactly two non-isomorphic limit models in every cardinal $\lambda\ge\aleph_1$ of Galois stability. In the adjacent class of all $R$-modules with ordinary embeddings the analogous spectrum is now fully understood — Mazari-Armida showed it equals $n+1$ exactly when $R$ is left $(<\aleph_n)$-noetherian but not $(<\aleph_{n-1})$-noetherian (a ring is left $(<\kappa)$-noetherian if every left ideal is generated by fewer than $\kappa$ elements), and that every infinite number of limit models can be realized by a suitable ring — which sharpens, but does not answer, the pure-embedding question.

Problem Statement

Under the source paper's standing hypothesis — $R$ an associative ring with unity, $T$ a (not necessarily complete) first-order theory with an infinite model extending the theory of left $R$-modules, and $\mathbf{K}^T=(\operatorname{Mod}(T),\leq_{\mathrm{pp}})$ with the joint embedding and amalgamation properties, where $\leq_{\mathrm{pp}}$ is the pure-submodule relation and $|T|=|R|+\aleph_0$ — determine the spectrum of limit models of $\mathbf{K}^T$ (Question 4.12 of arXiv:1903.00414, the preprint version of Kucera–Mazari-Armida 2020):

(1) Given a cardinal $\lambda$ with $\mathbf{K}^T$ $\lambda$-Galois-stable (so that $(\lambda,\alpha)$-limit models exist for all limit ordinals $\alpha<\lambda^+$), how many non-isomorphic limit models of cardinality $\lambda$ are there for a given $\mathbf{K}^T$, as a function of $\lambda$, the ring $R$ and the theory $T$? This includes the unresolved intermediate-cofinality regime $\omega<\operatorname{cf}(\alpha)<|T|^+$, where neither the pure-injectivity and uniqueness theorem for chains of cofinality at least $|T|^+$ nor the countable-cofinality description applies, and the question of whether the number can exceed two in any pure-embedding class of this form.

(2) Since the number of non-isomorphic limit models of cardinality $\lambda$ is bounded by one plus the number of infinite regular cardinals at most $|T|$ (Remark 4.11 of the source), is it always possible to find $R$ and $T$ satisfying the hypothesis such that $\mathbf{K}^T$ has this maximum number of non-isomorphic limit models?

The countable-ring case is already settled (eventually a unique limit model under Galois-superstability, otherwise exactly two), so the problem is substantive for uncountable $|T|$; answers are expected at this stated generality, not only for countable rings or complete theories.

References

1. Thomas G. Kucera and Marcos Mazari-Armida, On universal modules with pure embeddings, Mathematical Logic Quarterly 66(4) (2020) 395–408. DOI: 10.1002/malq.201900058. https://doi.org/10.1002/malq.201900058 (preprint with the question numbered 4.12: arXiv:1903.00414, https://arxiv.org/abs/1903.00414)
2. Marcos Mazari-Armida, Algebraic description of limit models in classes of abelian groups, Annals of Pure and Applied Logic 171(1) (2020) 102723. DOI: 10.1016/j.apal.2019.102723. https://doi.org/10.1016/j.apal.2019.102723
3. Marcos Mazari-Armida, Some stable non-elementary classes of modules, The Journal of Symbolic Logic 88(1) (2023) 93–117. DOI: 10.1017/jsl.2021.68. https://doi.org/10.1017/jsl.2021.68 (preprint: arXiv:2010.02918, https://arxiv.org/abs/2010.02918)
4. Marcos Mazari-Armida, A note on torsion modules with pure embeddings, Notre Dame Journal of Formal Logic 64(4) (2023). DOI: 10.1215/00294527-2024-0003. https://doi.org/10.1215/00294527-2024-0003 (preprint: arXiv:2104.10160, https://arxiv.org/abs/2104.10160)
5. Will Boney and Monica M. VanDieren, Limit models in strictly stable abstract elementary classes, Mathematical Logic Quarterly 70 (2024) 438–453. DOI: 10.1002/malq.202200075. https://doi.org/10.1002/malq.202200075 (preprint: arXiv:1508.04717, https://arxiv.org/abs/1508.04717)
6. Marcos Mazari-Armida, On limit models and parametrized Noetherian rings, Journal of Algebra 669 (2025) 58–74. DOI: 10.1016/j.jalgebra.2025.02.006. https://doi.org/10.1016/j.jalgebra.2025.02.006 (preprint: arXiv:2405.20214, https://arxiv.org/abs/2405.20214)
7. Jeremy Beard and Marcos Mazari-Armida, On the spectrum of limit models, Annals of Pure and Applied Logic 176(10) (2025) 103647. DOI: 10.1016/j.apal.2025.103647. https://doi.org/10.1016/j.apal.2025.103647 (preprint: arXiv:2503.11605, https://arxiv.org/abs/2503.11605)
8. Jeremy Beard, The spectrum of limit models in a first order setting, preprint (2025). arXiv:2510.03459. https://arxiv.org/abs/2510.03459
9. Will Boney, A module-theoretic introduction to abstract elementary classes, preprint (2025). arXiv:2503.10528. https://arxiv.org/abs/2503.10528