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Transactions of the American Mathematical Society

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Stable range one for rings with many idempotents


Authors: Victor P. Camillo and Hua-Ping Yu
Journal: Trans. Amer. Math. Soc. 347 (1995), 3141-3147
MSC: Primary 16D70; Secondary 16U50, 19B10
DOI: https://doi.org/10.1090/S0002-9947-1995-1277100-2
MathSciNet review: 1277100
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Abstract: An associative ring $ R$ is said to have stable range $ 1$ if for any $ a$, $ b \in R$ satisfying $ aR + bR = R$, there exists $ y \in R$ such that $ a + by$ by is a unit. The purpose of this note is to prove the following facts. Theorem $ 3$: An exchange ring $ R$ has stable range $ 1$ if and only if every regular element of $ R$ is unit-regular. Theorem $ 5$: If $ R$ is a strongly $ \pi $-regular ring with the property that all powers of every regular element are regular, then $ R$ has stable range $ 1$. The latter generalizes a recent result of Goodearl and Menal [$ 5$].


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1995-1277100-2
Keywords: Stable range one, exchange ring, strongly $ \pi $-regular ring
Article copyright: © Copyright 1995 American Mathematical Society

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