2024/04/20 by Khurana, Dinesh, Lam, T. Y.
#FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2404.13251
A ring element a∈ R is said to be of \it right stable range one\/ if, for any t∈ R, aR+tR=R implies that a+t b is a unit in R for some b∈ R. Similarly, a∈ 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'∈ R. In the last two decades, it has often been speculated that these two notions are actually the same for any a∈ 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∈ 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 [KL3]. 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.