Weakly relative quasi-injective modules

Abstract EN

Let R be a commutative ring with unity and let M, N be unitary R-modules.

In this research, we give generalizations for the concepts: weakly relative injectivity, relative tightness and weakly injectivity of modules.

We call M weakly N-quasi-injective, if for each f  Hom(N,  ) there exists a submodule X of  such that f (N)  X ≈ M , where  is the quasi-injective hull of M.

And we call M N-quasi-tight, if every quotient N / K of N which embeds in  embeds in M.

While we call M weakly quasi-injective if M is weakly N-quasiinjective for every finitely generated R-module N.

M oreover, we generalize some properties of weakly N-injective, N-tight and weakly injective modules to weakly N-quasi-injective, N-quasi-tight and weakly quasi-injective modules respectively.

The relations among these concepts are also studied.