@article{Rodbanjong2023, 
author = {Jaruwat Rodbanjong and Athipat Thamrongthanyalak},
title = {Characterizations of modules definable in o-minimal structures},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {6},
pages = {13088-13095},
keywords = {o-minimality, definable group, definable ring, definable module, manifold topology},
url = {https://www.sciopen.com/article/10.3934/math.2023660},
doi = {10.3934/math.2023660},
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 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        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        M   is definably connected with respect to the unique definable    S-module manifold topology.}
}