Rickart Modules Relative to Goldie Torsion Theory

TL;DR

This paper introduces Goldie Rickart modules, a new class of modules characterized by endomorphism properties related to the second singular submodule, and explores their properties and connections to certain ring classes.

Contribution

It defines Goldie Rickart modules using endomorphisms and characterizes their properties, linking them to semisimple and $ ext{Sigma-}t$-extending rings.

Findings

Goldie Rickart modules are characterized by endomorphisms with inverse images of the second singular submodule being direct summands.

Semisimple rings admit characterizations via Goldie Rickart modules.

Right $ ext{Sigma-}t$-extending rings are also characterized in terms of Goldie Rickart modules.

Abstract

Let $R$ be an arbitrary ring with identity and $M$ a right $R$-module with $S=$ End$_R(M)$. Let $Z_2(M)$ be the second singular submodule of $M$. In this paper, we define Goldie Rickart modules by utilizing the endomorphisms of a module. The module $M$ is called Goldie Rickart if for any $f\in S$, $f^{-1}(Z_2(M))$ is a direct summand of $M$. We provide several characterizations of Goldie Rickart modules and study their properties. Also we present that semisimple rings and right $\Sigma$-$t$-extending rings admit some characterizations in terms of Goldie Rickart modules.