A Generalization of Rickart Modules

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 Rickart modules. $\pi$-Rickart modules are also a dual notion of dual $\pi$-Rickart modules and extends that of generalized right principally projective rings to the module theoretic setting. 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 $r_M(f^n)=$ Ker$f^n=eM$. We obtain several results about generalized right principally projective rings by using $\pi$-Rickart modules. Moreover, we investigate relations between a $\pi$-Rickart module and its endomorphism ring.