Direct Injective Modules

Throughout this paper R will denote a ring with identity element and M a unitary right module over R. An R-module M is said to be direct injective if and only if given direct summand N of M with injection iN: N . M and a monomorphism g: N . M, there exists an endomorphism f of R-module M such that fg = iN. In this paper we investigate properties of direct injective modules, and obtain the following results on direct injective modules. (1) We establish the necessary and sufficient condition for a module to be direct injective. (2) We show that the answer on problem of Krull-Schmidt-Matlis is in the affirmative in case R-module M is extending direct injective. (3) We prove that extending direct injectivity of module implies same properties of its direct summands.