On Semiprime Goldie Modules

TL;DR

This paper generalizes Goldie's theorem to modules, establishing conditions under which a module is a semiprime Goldie module and linking module properties to the properties of its endomorphism ring.

Contribution

It introduces the concept of semiprime Goldie modules and characterizes them via endomorphism rings, extending classical ring theory results to modules.

Findings

Characterization of semiprime Goldie modules via endomorphism rings

Generalization of Goldie's theorem for modules

Equivalence of semiprime (prime) Goldie module properties and their endomorphism rings

Abstract

For an $R$-module $M$, projective in $\sigma[M]$ and satisfying ascending chain condition (ACC) on left annihilators, we introduce the concept of Goldie module. We also use the concept of semiprime module defined by Raggi et. al. in \cite{S} to give necessary and sufficient conditions for an $R$-module $M$, to be a semiprime Goldie module. This theorem is a generalization of Goldie's theorem for semiprime left Goldie rings. Moreover, we prove that $M$ is a semiprime (prime) Goldie module if and only if the ring $S=End_R(M)$ is a semiprime (prime) right Goldie ring. Also, we study the case when $M$ is a duo module.