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Finite $Σ$-Rickart modules

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 \emph{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 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. Also, 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 finite $Σ$-Rickart modules in terms of $M$-coherent modules. At the end, we study when $Σ$-Rickart modules and finite $Σ$-Rickart modules coincide. Examples which delineate the concepts and results are provided.