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.