Rickart Modules Relative to Goldie Torsion Theory

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.