Rmodule. We call the Rmodule M kerquasi-injective if for every monomorphism from into , where is a submodule of and is a quasi-injective hull of fN)(MQN)(MQ)M(QM and for every homomorphism from into gNM, there exists a homomorphism from Q into h)M(M such that . ghfkerker
It is clear that every quasi-injective module is kerquasi-injective, however the converse is false. Also every ker-injective module is kerquasi-injective, however the converse is false. In this paper we give some characterizations of kerquasi-injective modules, we also study some conditions under which a kerqausi-injective module becomes quasi-injective. For example, if a kerquasi-injective module is a finitely generated, then it is a quasi-injective. We ought to mention that we were not able to give an example of a kerquasi-injective module which is not quasi-injective and ker-injective.