A Generalization of Rickart Modules

Burcu Ungor; Sait Halıcıoglu; Abdullah Harmanci

paper

A Generalization of Rickart Modules

arXiv:1204.2343

Abstract

Let $R$ be an arbitrary ring with identity and $M$ a right $R$ -module with $S =$ End $_{R} (M)$ . In this paper we introduce $π$ -Rickart modules as a generalization of generalized right principally projective rings as well as that of Rickart modules. The module $M$ is called {\it $π$ -Rickart} if for any $f \in S$ , there exist $e^{2} = e \in S$ and a positive integer $n$ such that $r_{M} (f^{n}) = e M$ . We prove that several results of Rickart modules can be extended to $π$ -Rickart modules for this general settings, and investigate relations between a $π$ -Rickart module and its endomorphism ring.