Rings with each right ideal automorphism-invariant

TL;DR

This paper investigates rings where every right ideal is automorphism-invariant, characterizing their structure and properties, including decompositions, conditions for semisimplicity, and their behavior under various ring-theoretic conditions.

Contribution

It provides a comprehensive structural analysis of right automorphism-invariant rings, introducing the class of right $a$-rings and characterizing their properties and decompositions.

Findings

Right $a$-rings decompose into a semisimple and a square-free part.

Matrix rings over semisimple artinian rings are right $a$-rings.

Right $a$-rings are stably-finite and have specific regularity properties.

Abstract

In this paper, we study rings having the property that every right ideal is automorphism-invariant. Such rings are called right $a$-rings. It is shown that (1) a right $a$-ring is a direct sum of a square-full semisimple artinian ring and a right square-free ring, (2) a ring $R$ is semisimple artinian if and only if the matrix ring $\mathbb{M}_n(R)$ for some $n>1$ is a right $a$-ring, (3) every right $a$-ring is stably-finite, (4) a right $a$-ring is von Neumann regular if and only if it is semiprime, and (5) a prime right $a$-ring is simple artinian. We also describe the structure of an indecomposable right artinian right non-singular right $a$-ring as a triangular matrix ring of certain block matrices.