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- W2024280587 abstract "ABSTRACT A ring R has stable range one (respectively, weakly stable range one) provided whenever ax + b = 1 in R , there exists y ∈ R such that a + by is a two-sided (respectively, one-sided) inverse. In the first part of this article, we show that an exchange ring R has stable range one if and only if for every regular element x ∈ R there exist some unit-regular element w ∈ R and some idempotent e ∈ R such that x = ew if and only if whenever aR + bR = dR in R , there exists a unit-regular element w ∈ R such that a + bz = dw for some z ∈ R . These generalize a result of Canfell (1995 Canfell , M. J. ( 1995 ). Completions of diagrams by automorphisms and Bass’ first stable range condition . J. Alg. 176 : 480 – 513 . [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar], Theorem 2.9) and a special case of Chen (2000b Chen , H. ( 2000b ). On stable range conditions . Comm. Alg 28 ( 8 ): 3913 – 3924 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar], Theorem 4.2). In the second part, we prove that a module M satisfies the internal weak cancellation property if and only if every regular element of the endomorphism ring of M is one-sided unit-regular. We also extend a well known result on exchange rings with stable range one to exchange rings with weakly stable range one, by showing that an exchange ring R has weakly stable range one if and only if whenever in R , there exists a one-sided inverse u ∈ R such that au = ub ." @default.
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- W2024280587 date "2005-05-01" @default.
- W2024280587 modified "2023-09-26" @default.
- W2024280587 title "Unit-Regularity and Stable Range Conditions" @default.
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- W2024280587 doi "https://doi.org/10.1081/agb-200063337" @default.
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