On quasi pseudo-GP-injective rings and modules. (Q2016305)

A module \(M\) is called pseudo-injective if for any submodule \(N\) of \(M\), any monomorphism from \(N\) to \(M\) can be extended to an endomorphism of \(M\). Recently, Er, Singh and Srivastava have shown that these are precisely the modules which are invariant under automorphisms of their injective envelopes. As a generalization of pseudo-injective modules, this paper introduces the class of quasi pseudo-gp-injective modules. A module \(M\) is called quasi pseudo-gp-injective if for any nonzero endomorphism \(\alpha\) of \(M\), there exists a natural number \(n\) such that \(\alpha^n\neq 0\) and every monomorphism from \(\alpha^n(M)\) to \(M\) can be extended to an endomorphism of \(M\). The authors discuss various properties of such class of modules.