A Note on Rickart Modules

TL;DR

This paper explores Rickart modules, generalizing Baer modules and Rickart rings, providing characterizations, conditions for direct sums, and extending properties from rings to modules.

Contribution

It introduces a generalized notion of Rickart modules, offers new characterizations, and extends ring properties to module theory.

Findings

Characterizations of Rickart modules

Conditions under which direct sums are Rickart

Extension of ring properties to modules

Abstract

We study the notion of Rickart property in a general module theoretic setting as a generalization to the concept of Baer modules and right Rickart rings. A module $M_{R}$ is called Rickart if the right annihilator in $M_{R}$ of each left principal ideal of $End_{R}(M)$ is generated by an idempotent. Characterizations of Rickart modules are given. We give sufficient conditions for direct sums of Rickart modules to be Rickart. We extend some useful results of right Rickart rings to the theory of Rickart modules.