  In this article, we study the notion of a finite $\Sigma$-Rickart module, as
a module theoretic analogue of a right semi-hereditary ring. A module $M$ is
called \emph{finite $\Sigma$-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 $\Sigma$-Rickart module are finite $\Sigma$-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 $\Sigma$-Rickart module in terms of its endomorphism ring. In
addition, we introduce $M$-coherent modules and provide a characterization of
finite $\Sigma$-Rickart modules in terms of $M$-coherent modules. At the end,
we study when $\Sigma$-Rickart modules and finite $\Sigma$-Rickart modules
coincide. Examples which delineate the concepts and results are provided.
