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