Modules which are coinvariant under automorphisms of their projective covers

TL;DR

This paper investigates modules that remain unchanged under automorphisms of their projective covers, providing new proofs and characterizations, especially over right perfect rings, of when modules are coinvariant under such automorphisms.

Contribution

It offers a new, more conceptual proof for invariance under automorphisms of injective envelopes and characterizes coinvariance under automorphisms of projective covers over right perfect rings.

Findings

Modules invariant under automorphisms of their injective envelope have extension properties.

Modules coinvariant under automorphisms of their projective cover can be characterized by lifting epimorphisms.

The paper provides a dual perspective linking invariance and coinvariance properties.

Abstract

In this paper we study modules coinvariant under automorphisms of their projective covers. We first provide an alternative, and in fact, a more succinct and conceptual proof for the result that a module $M$ is invariant under automorphisms of its injective envelope if and only if given any submodule $N$ of $M$, any monomorphism $f:N\rightarrow M$ can be extended to an endomorphism of $M$ and then, as a dual of it, we show that over a right perfect ring, a module $M$ is coinvariant under automorphisms of its projective cover if and only if for every submodule $N$ of $M$, any epimorphism $\varphi: M\rightarrow M/N$ can be lifted to an endomorphism of $M$.