关于弱幺正则环和强稳定环

摘要

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.%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.