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Model Theory and Modules
, 1988
"... The modeltheoretic investigation of modules has led to ideas, techniques and results which are of algebraic interest, irrespective of their modeltheoretic significance. It is these aspects that I will discuss in this article, although I will make some comments on the model theory of modules per se ..."
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Cited by 64 (20 self)
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The modeltheoretic investigation of modules has led to ideas, techniques and results which are of algebraic interest, irrespective of their modeltheoretic significance. It is these aspects that I will discuss in this article, although I will make some comments on the model theory of modules per se. Our default is that the term “module ” will mean (unital) right module over a ring (associative with 1) R. The category of such modules is denoted ModR, the full subcategory of finitely presented modules will be denoted modR, the
VAUGHT’S CONJECTURE FOR MODULES OVER A
"... It is proved that Vaught’s conjecture is true for modules over an arbitrary countable Dedekind prime ring. It follows from the structural result that every module with few types over a countable Dedekind prime ring is ωstable. Vaught’s conjecture says that for any countable (complete) firstorder t ..."
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It is proved that Vaught’s conjecture is true for modules over an arbitrary countable Dedekind prime ring. It follows from the structural result that every module with few types over a countable Dedekind prime ring is ωstable. Vaught’s conjecture says that for any countable (complete) firstorder theory T, the number of nonisomorphic countable models of T is either countable or 2 ω, where ω is the first infinite cardinal. Vaught’s conjecture for ωstable theories of modules was proved by Garavaglia [6, Theorem 6]. Buechler proved that Vaught’s conjecture is correct for modules of Urank 1 [2]. It has been shown that Vaught’s conjecture for finite Urank may be reduced to the case of certain abelian structures, and these may be turned into modules [10, p. 167]. Baldwin and McKenzie proved that Vaught’s conjecture is true for the theory of all modules over a countable ring [1]. The natural approach in this topic is to fix some suitable class of rings for consideration, allowing the complete theory of modules to be arbitrary. Ziegler [15] checked Vaught’s conjecture for complete theories of modules over a countable