Characterizations of modules definable in o-minimal structures

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