Additive Unit Representations in Endomorphism Rings and an Extension of a result of Dickson and Fuller

TL;DR

This paper extends a known result by showing that automorphism-invariant modules over certain algebras are quasi-injective, using additive unit structures of endomorphism rings to simplify the proof.

Contribution

It generalizes and simplifies the proof that automorphism-invariant modules over algebras with more than two elements are quasi-injective.

Findings

Automorphism-invariant modules over such algebras are quasi-injective.

The proof is simplified by analyzing additive units in endomorphism rings.

The result extends previous work by Dickson and Fuller.

Abstract

A module is called automorphism-invariant if it is invariant under any automorphism of its injective hull. Dickson and Fuller have shown that if $R$ is a finite-dimensional algebra over a field $\mathbb F$ with more than two elements then an indecomposable automorphism-invariant right $R$-module must be quasi-injective. In this note, we extend and simplify the proof of this result by showing that any automorphism-invariant module over an algebra over a field with more than two elements is quasi-injective. Our proof is based on the study of the additive unit structure of endomorphism rings.