Definable Version of Wedderburn–Artin Theorem in O-Minimal Structures

Here we work in an arbitrary o-minimal expansion of a divisible ordered abelian group. We say that a definable ring is definably semiprime if squares of nontrivial two-sided ideals definable in the expansion are nontrivial. We prove a definable version of Wedderburn–Artin theorem and give a characterization of definably semiprime rings.