Dual $\pi$-Rickart Modules

TL;DR

This paper introduces dual $ ext{ extpi}$-Rickart modules, generalizing $ ext{ extpi}$-regular rings and dual Rickart modules, and explores their properties and relationships with endomorphism rings.

Contribution

It defines dual $ ext{ extpi}$-Rickart modules, extending existing concepts, and investigates their properties and connections to endomorphism rings.

Findings

Dual $ ext{ extpi}$-Rickart modules generalize dual Rickart modules.

Results from dual Rickart modules extend to dual $ ext{ extpi}$-Rickart modules.

Analyzes the relationship between dual $ ext{ extpi}$-Rickart modules and their endomorphism rings.

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