Rings of Invariant Module Type and Automorphism-Invariant Modules

TL;DR

This paper investigates automorphism-invariant modules over rings, extending known results from finite-dimensional algebras to right artinian rings, and characterizes indecomposable right artinian rings of automorphism-invariant type.

Contribution

It generalizes the characterization of rings where all indecomposable modules are automorphism-invariant to right artinian rings.

Findings

Fails to hold over fields with two elements.

Extends Dickson and Fuller's results to right artinian rings.

Provides a complete characterization of indecomposable right artinian rings of RAI-type.

Abstract

A module is called automorphism-invariant if it is invariant under any automorphism of its injective hull. In [Algebras for which every indecomposable right module is invariant in its injective envelope, Pacific J. Math., vol. 31, no. 3 (1969), 655-658] Dickson and Fuller had 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 paper we show that this result fails to hold if $\mathbb F$ is a field with two elements. Dickson and Fuller had further shown that if $R$ is a finite-dimensional algebra over a field $\mathbb F$ with more than two elements, then $R$ is of right invariant module type if and only if every indecomposable right $R$-module is automorphism-invariant. We extend the result of Dickson and Fuller to any right artinian ring. A ring $R$ is said to be of right automorphism-invariant type (in short, RAI-type) if every finitely generated indecomposable right $R$-module is automorphism-invariant. In this paper we completely characterize an indecomposable right artinian ring of RAI-type.