Finite Σ-Rickart modules

After hereditary rings were defined, as a natural generalization, semi-hereditary rings were introduced. In this article, we study the notion of a finite Σ-Rickart module, as a module theoretic analogue of a right semi-hereditary ring. A module M is called finite Σ-Rickart if every finite direct sum of copies of M is a Rickart module. It is shown that any direct summand and any finite direct sum of copies of a finite Σ-Rickart module are finite Σ-Rickart modules. We also provide generalizations in a module theoretic setting of the most common results of semi-hereditary rings: a ring R is right semi-hereditary if and only if every finitely generated submodule of any projective right R-module is projective if and only if every factor module of any injective right R-module is f-injective. Further, we have a characterization of a finite Σ-Rickart module in terms of its endomorphism ring. In addition, we introduce M-coherent modules and provide a characterization of a finite Σ-Rickart module in terms of an M-coherent module. We prove that for any endoregular module M, HomR(M,A) is an f-injective right S-module for any module A, where S=EndR(M). Examples which delineate the concepts and results are provided.