MODULES AND RINGS IN WHICH EVERY COMPLEMENT IS ISOMORPHIC TO A SUMMAND

In this paper, we introduce a generalization of the well-known CS condition. We say an -module  is a -  if every complement is isomorphic to a summand. We prove that if  is a right CIS-ring right FGF ring, then  is a quasi-Frobenius ring, and if  is a right CIS-ring right CF ring, then  is a right artinian ring. New characterizations of quasi-Frobenius rings are provided by using CIS-rings. Moreover, many of the important propositions related to CS-rings are generalized to CIS-rings also presented.