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A ring element a ∈ R is said to be of right stable range one if, for any t ∈ R , a R + t R = 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 − a b is a unit in R iff so is 1 − b a. 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 [45]. 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. [ABSTRACT FROM AUTHOR]

The equation XA-BX = I has been well studied in ring theory, operator theory, linear algebra, and other branches of mathematics. In this paper, we show that, in the case where B2 = B, the study of XA - BX = I in a noncommutative ring R leads to several new ways to view and to work with the exchange (or "suitable") elements in R in the sense of Nicholson. For any exchange element A ∊ R, we show that the set of idempotents E ∊ R such that E ∊ RA and I - E ∊ R(I - A) is naturally parametrized by the roots of a certain left-right symmetric "exchange polynomial" associated with A. From the new viewpoints on exchange elements developed in this paper, the classes of clean and strongly clean elements in rings can also be better understood. [ABSTRACT FROM AUTHOR]