Weakly Relative Quasi-Injective Modules

Let R be a commutative ring with unity and let M, N be unitary R-modules. In thisresearch, we give generalizations for the concepts: weakly relative injectivity, relativetightness and weakly injectivity of modules. We call M weakly N-quasi-injective, if for each fє Hom(N, M ) there exists a submodule X of M such that f (N) subseteq X ≈ M , where M is thequasi-injective hull of M. And we call M N-quasi- tight, if every quotient N / K of N whichembeds in M embeds in M. While we call M weakly quasi- injective if M is weakly N-quasiinjectivefor every finitely generated R-module N.M oreover, we generalize some properties of weakly N-injective, N-tight and weaklyinjective modules to weakly N- quasi-injective, N-quasi-tight and weakly quasi-injectivemodules respectively. The relations among these concepts are also studied.