Ring Elements of Stable Range One

A ring element \,a∈ R\, is said to be of { right stable range one\/} if, for any \,t∈ R, $\,aR+tR=R\,$ implies that $\,a+t\,b\,$ is a unit in $\,R\,$ for some \,b∈ R. Similarly, \,a∈ R\, is said to be of { 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 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 [KL₃]. 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.