ON PRINCIPALLY QUASI-BAER MODULES

Abstract

Let $R$ be an arbitrary ring with identity and $M$ a right $R$-module with $S={\mathrm{End}}_R\left(M\right)$. In this paper, we introduce a class of modules that is a generalization of principally quasi-Baer rings and Baer modules. The module ${}_{S}M$ is called principally quasi-Baer if for any $m\in M,l_S\left(Sm\right)=Se$ for some $e^2=e\in S$. It is proved that (1) if ${}_{S}M$ is regular and semicommutative module or (2) if $M_R$ is principally semisimple and ${}_{S}M$ is abelian, then ${}_{S}M$ is a principally quasi-Baer module. The connection between a principally quasi-Baer module ${}_{S}M$ and polynomial extension, power series extension, Laurent polynomial extension, Laurent power series extension of ${}_{S}M$ is investigated.