Ring Elements of Stable Range One

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 $a+tb$ 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 $Ra+Rt=R$ implies that $a+b^{\prime }t$ is a unit in $R$ for some $b^{\prime }\in R$. 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 §\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.

31 pages