Rings characterized by their weakly-injective modules

Abstract

The notation in this paper will be standard and it may be found in [2] or [8]. Throughout the paper, the notation A ⊂' B will mean that A is an essential submodule of the module B. Given an arbitrary ring R and R-modules M and N, we say that M is weakly N-injective if and only if every map φ:N → E(M) from N into the injective hull E(M) of M may be written as a composition σ〫 , where :N→M and σ:M→E(M) is a monomorphism. This is equivalent to saying that for every map φ:N→E(M), there exists a submodule X of E(M), isomorphic to M, such that φ(N) is contained in X. In particular, M is weakly R-injective if and only if, for every x ∈ E(M), there exists X ⊂ E(M) such that x ∈ X ≌ M. We say that M is weakly-injective if and only if it is weakly N-innjective for every finitely generated module N. Clearly, M is weakly-injective if and only if, for every finitely generated submodule N of E(M), there exists X ⊂ E(M) such that N ⊂ X ≌ M.