# Automorphism-invariant non-singular rings and modules

## TL;DR

This paper characterizes automorphism-invariant non-singular rings and modules, establishing their structure and properties through equivalences involving regularity, injectivity, and Goldie conditions.

## Contribution

It provides new characterizations and structural descriptions of automorphism-invariant non-singular rings and modules, linking them to regularity and injectivity properties.

## Key findings

- A ring is automorphism-invariant non-singular iff it decomposes into specific regular and strongly regular components.
- Conditions under which direct sums of automorphism-invariant modules are injective or automorphism-invariant are established.
- The structure of rings with certain Goldie radical properties is characterized through module invariance and semiprimeness.

## Abstract

$\textbf{Theorem 1.2.}$ For a ring $A$, the following conditions are equivalent. $\textbf{1)}$ $A$ is a right automorphism-invariant right non-singular ring. $\textbf{2)}$ $A$ is a right automorphism-invariant regular ring. $\textbf{3)}$ $A=S\times T$, where $S$ is a right injective regular ring and $T$ is a strongly regular ring which contains all invertible elements of its maximal right ring of quotients. $\textbf{Theorem 1.5.}$ For a ring $A$ with right Goldie radical $G(A_A)$, the following conditions are equivalent. $\textbf{1)}$ $A/G(A_A)$ is a semiprime right Goldie ring. $\textbf{2)}$ Any direct sum of automorphism-invariant non-singular right $A$-modules is an automorphism-invariant module. $\textbf{3)}$ Any direct sum of automorphism-invariant non-singular right $A$-modules is an injective module.