关于强Clean一般环

本文介绍了强clean—般环的概念并将一些基本的结果推广到这个更广的环类．证明了强clean一般环的角落环和强π-正则一般环都是强clean的,还讨论了强clean一般环的扩张并且证明了满足条件J(I)=Q(I)的交换clean一般环的上三角矩阵环是强clean的．

Clean一般环的几点注记

环是指具有单位元的结合环,而一般环是指有或没有单位元的结合环.一般环I称为是强Clean的,如果I中每个元素a具有下述的形式a=e+q,其中e^2=e∈I,q∈Q(I)={q∈I|q+p+pq=p+q+pq=0对某个p∈I}且eq=qe.这一概念是Nicholson中强Clean环概念的真推广,强Clean一般环的刻画给出了。基于此,证明了强Clean一般环的单边理想也是强Clean的,并证明了如果I是强Clean一般环,那么,对于任意x∈I,I在x处的局部环I_x也是强Clean的,特别地,强Clean一般环的角落子环eRe总是强Clean的对于任意的e^2=e∈I.这推广了Chen中的主要结果.

Extensions of strongly π-regular general rings

The concept of the strongly π-regular general ring （with or without unity） is introduced and some extensions of strongly π-regular general rings are considered. Two equivalent characterizations on strongly π- regular general rings are provided. It is shown that I is strongly π-regular if and only if, for each x ∈I, x^n =x^n＋1y = zx^n＋1 for n ≥ 1 and y, z ∈ I if and only if every element of I is strongly π-regular. It is also proved that every upper triangular matrix general ring over a strongly π-regular general ring is strongly π-regular and the trivial extension of the strongly π-regular general ring is strongly clean.