SOME STABLE NON-ELEMENTARY CLASSES OF MODULES

被引:5
作者
Mazari-Armida, Marcos [1 ,2 ]
机构
[1] Carnegie Mellon Univ, Dept Math Sci, Pittsburgh, PA 15213 USA
[2] Univ Colorado, Dept Math, Boulder, CO 80309 USA
关键词
stability; abstract elementary classes; superstability; noetherian rings; pure-semisimple rings; universal models; PURE; RINGS; EXTENSIONS;
D O I
10.1017/jsl.2021.68
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
Fisher [10] and Baur [6] showed independently in the seventies that if T is a complete first-order theory extending the theory of modules, then the class of models of T with pure embeddings is stable. In [25, 2.12], it is asked if the same is true for any abstract elementary class (K,<=(p)) such that K is a class of modules and <=(p) is the pure submodule relation. In this paper we give some instances where this is true: THEOREM. Assume R is an associative ring with unity. Let (K,<=(p)) be an AEC such that K subset of R-Mod and K is closed under finite direct sums, then: If K is closed under pure-injective envelopes, then K is lambda-stable for every lambda = LS(K) such that lambda(vertical bar R vertical bar+N0) = lambda If K is closed under pure submodules and pure epimorphic images, then K is lambda-stable for every lambda such that lambda(vertical bar R vertical bar+N0) = lambda. Assume R is Von Neumann regular. If K is closed under submodules and has arbitrarily large models, then K is lambda-stable for every lambda such that lambda(vertical bar R vertical bar+N0) = lambda. As an application of these results we give new characterizations of noetherian rings, pure-semisimple rings, Dedekind domains, and fields via superstability. Moreover, we show how these results can be used to show a link between being good in the stability hierarchy and being good in the axiomatizability hierarchy. Another application is the existence of universal models with respect to pure embeddings in several classes of modules. Among them, the class of flat modules and the class of s-torsion modules.
引用
收藏
页码:93 / 117
页数:25
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