Clean一般环的几点注记

A NOTE ON CLEAN GENERAL RINGS

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.

By a ring we mean an associative ring with unity and a general ring we mean a ring with or without unity. A general ring I is called strongly clean if each element a of I has the form a=e＋q where e^2=e ∈ I, q∈Q（I） = q∈ I： q＋p＋qp = p＋q＋pq =0 for some p∈I with eq = qe. This notion is a proper generalization of that of a strongly clean ring in Nicholson. A characterization of a strongly clean general ring is given. Based on this, it is proved that any one-sided ideal of a strongly clean general ring is also strongly clean and that if I is a strongly clean ring, then so is the local ring Ix of I at x. In particular, the comer subring eRe of a strongly clean general ring is strongly clean for any e2 = e, which generalizes the main result in Chen.