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 pi-Rickart if for any f is an element of S, there exist e(2) = e is an element of S and a positive integer n such that r(M) (f(n)) = Kerf(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.