Modules which are subisomorphic to quasi-injective modules

Abstract EN

Let R be a commutative ring with identity and let M be a unitary left 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.