Submission date: 7 February 2023

Abstract:

Let M be an o-minimal expansion of a densely linearly ordered set and (S,+,∙,0_S,1_S) be a ring definable in M. In this article, we develop two techniques for the study of characterizations of S-modules definable in M. The first technique is an algebraic technique. More precisely, we show that every S-module definable in M is finitely generated. For the other technique, we prove that if S is an infinite ring without zero divisors, every S-module definable in M admits a unique definable S-module manifold topology. As consequences, we obtain the following: (1) if S is finite, then a module A is isomorphic to an S-module definable in M if and only if A is finite; (2) if S is an infinite ring without zero divisors, then a module A is isomorphic to an S-module definable in M if and only if A is a finite dimensional free module over S; and (3) if S is an infinite ring without zero divisors, then every S-module definable in M is connected with respect to the unique definable S-module manifold topology.

Mathematics Subject Classification: Primary 03C64; Secondary 16D10, 16D40, 16W80, 18F15

Keywords and phrases: o-minimality, definable group, definable ring, definable module, manifold topology

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