On the structure of Goldie Modules

TL;DR

This paper explores the structure of semiprime Goldie modules, their decompositions, and endomorphism rings, extending classical ring theory results to module categories and establishing new relationships among various module classes.

Contribution

It provides a new characterization of semiprime Goldie modules via decompositions and relates these modules to $QI$-modules and co-semisimple modules.

Findings

Decomposition of the $M$-injective hull in terms of minimal prime submodules.

Characterization of semiprime Goldie modules in $ ext{Z-Mod}$.

Relations among semiprime Goldie modules, $QI$-modules, and co-semisimple modules.

Abstract

Given a semiprime Goldie module $M$ projective in $\sigma[M]$ we study decompositions on its $M$-injective hull $\hat{M}$ in terms of the minimal prime in $M$ submodules. With this, we characterize the semiprime Goldie modules in $\mathbb{Z}$-Mod and make a decomposition of the endomorphism ring of $\hat{M}$. Also, we investigate the relations among semiprime Goldie modules, $QI$-modules and co-semisimple modules extending results on left $QI$-rings and $V$-rings.