On essentially Π-injective modules and rings

In this paper, we study modules having the property that are invariant under some idempotent endomorphisms of its injective envelope. Such modules are called essentially π-injective. It is shown that (1) M is essentially π-injective iff for any essentially finite direct summand X1 of M and any submodule X2 of M with X1∩X2=0, there exists a direct summand X0 of M containing X2 such that M=X1⊕X0, (2) M is essentially π-injective iff M is an ef-extending right R-module and for any decomposition M=M1⊕M2 with M1 essentially finite, M1 and M2 are relatively injective, (3) if M is essentially π-injective and R satisfies ACC on right ideals of the form r(m), m∈M, then M is a direct sum of uniform submodules. We also describe rings via essentially π-injective modules. It is shown that R is a semisimple artinian ring iff the direct sum of any two essentially π-injective right R-modules is essentially π-injective.