Ring elements of stable range one

Abstract

A ring element a∈R is said to be of right stable range one if, for any t∈R, aR+tR=R implies that a+tb is a unit in R for some b∈R. Similarly, a∈R is said to be of left stable range one if Ra+Rt=R implies that a+b′t is a unit in R for some b′∈R. In the last two decades, it has often been speculated that these two notions are actually the same for any a∈R. In §3 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∈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 [46]. In §§4-5, a detailed study is offered for stable range one ring elements that are unit-regular or nilpotent, while §6 examines the behavior of stable range one elements via their classical Peirce decompositions. The paper ends with a more concrete §7 on integral matrices of stable range one, followed by a final §8 with a few open questions.