Ring Elements of Stable Range One

TL;DR

This paper proves the equivalence of right and left stable range one for ring elements, introduces a generalized Jacobson's lemma, and explores properties of such elements including their behavior in matrices and special cases.

Contribution

It establishes the symmetry of right and left stable range one, introduces a super Jacobson's lemma, and generalizes Sylvester's determinantal identity.

Findings

Right and left stable range one are equivalent for any ring element.

A new super Jacobson's lemma is proved, generalizing classical results.

Characterization of stable range one elements in matrices and special classes.

Abstract

A ring element $\,a\in R\,$ is said to be of {\it right stable range one\/} if, for any $\,t\in R$, $\,aR+tR=R\,$ implies that $\,a+t\,b\,$ is a unit in $\,R\,$ for some $\,b\in R$. Similarly, $\,a\in R\,$ is said to be of {\it left stable range one\/} if $\,R\,a+R\,t=R\,$ implies that $\,a+b't\,$ is a unit in $\,R\,$ for some $\,b'\in R$. In the last two decades, it has often been speculated that these two notions are actually the same for any $\,a\in R$. In \S3 of this paper, we will prove that this is indeed the case. The key to the proof of this new symmetry result is a certain ``Super Jacobson's Lemma'', which generalizes Jacobson's classical lemma stating that, for any $\,a,b\in R$, $\,1-ab\,$ is a unit in $\,R\,$ iff so is $\,1-ba$. Our proof for the symmetry result above has led to a new generalization of a classical determinantal identity of Sylvester, which will be published separately in [KL$_3$]. In \S\S4-5, a detailed study is offered for stable range one ring elements that are unit-regular or nilpotent, while \S6 examines the behavior of stable range one elements via their classical Peirce decompositions. The paper ends with a more concrete \S7 on integral matrices of stable range one, followed by a final \S8 with a few open questions.