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In this paper, we study modules whose endomorphism rings are right Rickart (or right p.p.) rings, which we call R-endoRickart modules. We provide some characterizations of R-endoRickart modules. Some classes of rings are characterized in terms of R-endoRickart modules. We prove that an R-endoRickart module with no innite set of nonzero orthogonal idempotents in its endomorphism ring is precisely an endoBaer module. We show that a direct summand of an R-endoRickart modules inherits the property, while a direct sum of R-endoRickart modules does not. Necessary and sucient conditions for a nite direct sum of R-endoRickart modules to be an R-endoRickart module are provided.