  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.
