Vaught’s Conjecture for Modules over a Dedekind Prime Ring

Vaught’s conjecture says that for any countable (complete) first-order theory T, the number of non-isomorphic countable models of T is either countable or 2, where ω is the first infinite cardinal. Vaught’s conjecture for ω-stable theories of modules was proved by Garavaglia [6, Theorem 6]. Buechler proved that Vaught’s conjecture is correct for modules of U-rank 1 [2]. It has been shown that Vaught’s conjecture for finite U-rank 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 commutative Dedekind domain, and Puninskaya [12] verified the similar result for modules over a countable serial ring. In both cases, the main point is to reduce the problem to the ω-stable case. In this paper we prove that Vaught’s conjecture is true for complete theories of modules over a countable (noncommutative) Dedekind prime ring by showing that every module with few types over a countable Dedekind prime ring is ω-stable. This result covers a large class of interesting noncommutative examples, such as the first Weyl algebra over a countable field of characteristic zero. The main technical tool which we use is a variant of the Zariski topology over a ring. This consideration is motivated by the investigation of ω-stable modules over a Pru$ fer ring [3]. When this topology is discrete, continuum many types are obtained by using Lemma 1.1 of [7]. Using some results of Prest [11], we obtain as a corollary that Vaught’s conjecture is true for modules over a tame hereditary finite-dimensional algebra over a countable infinite field.