Dual automorphism-invariant modules

A module M is called an automorphism-invariant module if every isomorphism between two essential submodules of M extends to an automorphism of M. This paper introduces the notion of dual of such modules. We call a module M to be a dual automorphism-invariant module if whenever K1 and K2 are small submodules of M, then any epimorphism η:M/K1→M/K2 with small kernel lifts to an endomorphism φ of M. In this paper we give various examples of dual automorphism-invariant module and study its properties. In particular, we study abelian groups and prove that dual automorphism-invariant abelian groups must be reduced. It is shown that over a right perfect ring R, a lifting right R-module M is dual automorphism-invariant if and only if M is quasi-projective.