Extensions of square stable range one

TL;DR

This paper characterizes when exchange ideals in rings are square stable, linking this property to conditions on elements related to the Jacobson radical and strong regularity.

Contribution

It provides a characterization of square stable exchange ideals in rings through conditions involving the Jacobson radical and strong regularity.

Findings

An exchange ideal is square stable iff certain radical conditions hold.

Square stability is equivalent to all regular elements being strongly regular.

The paper establishes a clear criterion connecting radical properties and stability.

Abstract

An ideal $I$ of a ring $R$ is square stable if $aR+bR=R$ with $a\in I$ and $b\in R$ implies that $a^2+by$ is invertible in $I$ for some $y\in I$. We prove that an exchange ideal $I$ of a ring $R$ is square stable if and only if for any $a\in I$, $a^2\ in J(R)$ implies that $a\in J(R)$, if and only if every regular element in $I$ is strongly regular.