1. Ring elements of stable range one.
- Author
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Khurana, Dinesh and Lam, T.Y.
- Subjects
- *
OPEN-ended questions , *SYMMETRY - Abstract
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]
- Published
- 2024
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