南京大学学报（数学半年刊）
南京大學學報（數學半年刊）
남경대학학보（수학반년간）
JOURNAL OF NANJING UNIVERSITY MATHEMATICAL BIQUARTERLY
2004年
2期
194-200
,共7页
弱幺正则环%强稳定环%K1-群
弱幺正則環%彊穩定環%K1-群
약요정칙배%강은정배%K1-군
weakly unit-regular rings%strongly stable rings%K1-groups
在本文中我们引入了弱幺正则环的概念,证明这类环是幺正则环和半局部环的自然推广.另外,我们还证明了下面两个结果:(1)环R是弱幺正则的当且仅当Mn(R)(n≥1)是弱幺正则环;(2)假设R是一个环且使得R/J(R)是正则环.那么R是弱正则环当且仅当对任意ax+b=1存在v∈R和一个左可逆元u∈R使得au+bv=1以及当且仅当对任意x∈R存在一个左可逆元u∈R以及y∈J(R)使得x+y=xux.
在本文中我們引入瞭弱幺正則環的概唸,證明這類環是幺正則環和半跼部環的自然推廣.另外,我們還證明瞭下麵兩箇結果:(1)環R是弱幺正則的噹且僅噹Mn(R)(n≥1)是弱幺正則環;(2)假設R是一箇環且使得R/J(R)是正則環.那麽R是弱正則環噹且僅噹對任意ax+b=1存在v∈R和一箇左可逆元u∈R使得au+bv=1以及噹且僅噹對任意x∈R存在一箇左可逆元u∈R以及y∈J(R)使得x+y=xux.
재본문중아문인입료약요정칙배적개념,증명저류배시요정칙배화반국부배적자연추엄.령외,아문환증명료하면량개결과:(1)배R시약요정칙적당차부당Mn(R)(n≥1)시약요정칙배;(2)가설R시일개배차사득R/J(R)시정칙배.나요R시약정칙배당차부당대임의ax+b=1존재v∈R화일개좌가역원u∈R사득au+bv=1이급당차부당대임의x∈R존재일개좌가역원u∈R이급y∈J(R)사득x+y=xux.
In this paper, we introduce the concepts of weakly unit-regular rings and prove that these rings are a natural generalization of unit-regular rings and semilocal rings. In addition, we obtain the following results: (1) A ring R is weakly unit-regular if and only if so is the n×n matrix ring Mn(R) of R for any n≥1; (2) Let R be a ring such that R/J(R) is regular.Then R is a weakly unit-regular ring if and only if whenever ax+b=1 in R, there exist a left invertible u∈R and v∈R such that au+bv= 1, and if and only if for any x∈R there exist a left invertible u∈ R and y∈J(R) such that x+y=xux.
