Hereditary rings have been extensively investigated in the literature after Kaplansky introduced them in the earliest 50's. In this paper, we study the notion of a Sigma-Rickart module by utilizing the endomorphism ring of a module and using the recent notion of a Rickart module, as a module theoretic analogue of a right hereditary ring. A module M is called Sigma-Rickart if every direct sum of copies of M is Rickart. It is shown that any direct summand and any direct sum of copies of a Sigma-Rickart module are Sigma-Rickart modules. We also provide generalizations in a module theoretic setting of the most common results of hereditary rings: a ring R is right hereditary if and only if every submodule of any projective right R-module is projective if and only if every factor module of any injective right R-module is injective. Also, we have a characterization of a finitely generated Sigma-Rickart module in terms of its endomorphism ring. Examples which delineate the concepts and results are provided.
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