We initiate a systematic investigation of the abstract elementary classes that have amalgamation, satisfy tameness, and are stable in some cardinal. Assuming the singular cardinal hypothesis, we prove a full characterization of the stability cardinals, and connect the stability spectrum with the behavior of saturated models.We deduce that if a class is stable on a tail of cardinals, then it has no long splitting chains. This indicates that there is a clear notion of superstability in this framework.We also present an (...) application to homogeneous model theory: for [Formula: see text] a homogeneous diagram in a first-order theory [Formula: see text], if [Formula: see text] is both stable in [Formula: see text] and categorical in [Formula: see text] then [Formula: see text] is stable in all [Formula: see text]. (shrink)

In this paper we study abstract elementary classes of modules. We give several characterizations of when the class of modules A with is abstract elementary class with respect to the notion that M1 is a strong submodel M2 if the quotient remains in the given class.

We introduce tame abstract elementary classes as a generalization of all cases of abstract elementary classes that are known to permit development of stability-like theory. In this paper, we explore stability results in this new context. We assume that [Formula: see text] is a tame abstract elementary class satisfying the amalgamation property with no maximal model. The main results include:. Theorem 0.1. Suppose that [Formula: see text] is not only tame, but [Formula: see text]-tame. If [Formula: see text] and [Formula: (...) see text] is Galois stable in μ, then [Formula: see text], where [Formula: see text] is a relative of κ from first order logic. [Formula: see text] is the Hanf number of the class [Formula: see text]. It is known that [Formula: see text]. The theorem generalizes a result from [17]. It is used to prove both the existence of Morley sequences for non-splitting and the following initial step towards a stability spectrum theorem for tame classes:. Theorem 0.2. If [Formula: see text] is Galois-stable in some [Formula: see text], then [Formula: see text] is stable in every κ with κμ=κ. For example, under GCH we have that [Formula: see text] Galois-stable in μ implies that [Formula: see text] is Galois-stable in μ+n for all n < ω. (shrink)

We deal with the existence of universal members in a given cardinality for several classes. First, we deal with classes of abelian groups, specifically with the existence of universal members in cardinalities which are strong limit singular of countable cofinality or λ=λℵ0. We use versions of being reduced—replacing Q by a subring —and get quite accurate results for the existence of universals in a cardinal, for embeddings and for pure embeddings. Second, we deal with the oak property, a property of (...) complete first-order theories sufficient for the nonexistence of universal models under suitable cardinal assumptions. Third, we prove that the oak property holds for the class of groups and deals more with the existence of universals. (shrink)

We characterize rings over which every cotorsion module is pure injective in terms of certain descending chain conditions and the Ziegler spectrum, which renders the classes of von Neumann regular rings and of pure semisimple rings as two possible extremes. As preparation, descriptions of pure projective and Mittag–Leffler preenvelopes with respect to so-called definable subcategories and of pure generation for such are derived, which may be of interest on their own. Infinitary axiomatizations lead to coherence results previously known for the (...) special case of flat modules. Along with pseudoflat modules we introduce quasiflat modules, which arise naturally in the model-theoretic and the category-theoretic contexts. (shrink)

We show that certain classes of modules have universal models with respect to pure embeddings: Let R be a ring, T a first‐order theory with an infinite model extending the theory of R‐modules and (where ⩽pp stands for “pure submodule”). Assume has the joint embedding and amalgamation properties. If or, then has a universal model of cardinality λ. As a special case, we get a recent result of Shelah [28, 1.2] concerning the existence of universal reduced torsion‐free abelian groups with (...) respect to pure embeddings.We begin the study of limit models for classes of R‐modules with joint embedding and amalgamation. We show that limit models with chains of long cofinality are pure‐injective and we characterize limit models with chains of countable cofinality. This can be used to answer [18, Question 4.25].As this paper is aimed at model theorists and algebraists an effort was made to provide the background for both. (shrink)