Dual automorphism-invariant modules

TL;DR

This paper introduces the concept of dual automorphism-invariant modules, explores their properties, and characterizes them in the context of abelian groups and modules over right perfect rings.

Contribution

It defines dual automorphism-invariant modules, provides examples, and establishes their properties and characterizations in specific algebraic contexts.

Findings

Dual automorphism-invariant abelian groups are reduced.

Over right perfect rings, dual automorphism-invariant modules are equivalent to quasi-projective modules.

Abstract

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 $K_1$ and $K_2$ are small submodules of $M$, then any epimorphism $\eta:M/K_1\rightarrow M/K_2$ with small kernel lifts to an endomorphism $\varphi$ 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.