A Generalization of Rickart Modules

TL;DR

This paper introduces $ ext{ extpi}$-Rickart modules, a broad generalization of Rickart modules and related rings, extending key properties and exploring their relationship with endomorphism rings.

Contribution

The paper defines $ ext{ extpi}$-Rickart modules, generalizing Rickart modules, and extends several properties and results to this new class, linking module and endomorphism ring structures.

Findings

$ ext{ extpi}$-Rickart modules generalize Rickart modules and generalized right principally projective rings.

Several properties of Rickart modules are extended to $ ext{ extpi}$-Rickart modules.

Relations between $ ext{ extpi}$-Rickart modules and their endomorphism rings are established.

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 $\pi$-Rickart modules as a generalization of generalized right principally projective rings as well as that of Rickart modules. The module $M$ is called {\it $\pi$-Rickart} if for any $f\in S$, there exist $e^2=e\in S$ and a positive integer $n$ such that $r_M(f^n)=eM$. We prove that several results of Rickart modules can be extended to $\pi$-Rickart modules for this general settings, and investigate relations between a $\pi$-Rickart module and its endomorphism ring.