  The Mitsch order is already known as a natural partial order for semigroups
and rings. The purpose of this paper is to further study of the Mitsch order on
modules by investigating basic properties via endomorphism rings. And so this
study also contribute to the results related to the orders on rings. As a
module theoretic analog of the Mitsch order, we show that this order is a
partial order on arbitrary modules. Among others, lattice properties of the
Mitsch order and the relations between the Mitsch order and the other
well-known orders, such as, the minus order, the Jones order, the direct sum
order and the space pre-order on modules are studied. In particular, we prove
that the minus order is the Mitsch order and we supply an example to show that
the converse does not hold in general.
