一般环的广义弱clean指数（英文）

Weakly Clean General Index of General Rings

一般环(未必有单位)中的元素a称为clean,若其可表为一个幂等元和一个Q(R)中元素之和;一般环I称为clean general环,若环I中元素都是clean的.受clean和弱clean指数概念的启发,我们给出对于一般环的一类新的指数——广义弱clean指数,给出该指数的一些性质,并且证明了一般环的广义弱clean指数为1时,该环为abelian环.进一步地,我们给出一般环的广义弱clean指数为2或3时环的性质刻画,得到了一些有关矩阵环的广义弱clean指数的性质.而对于一些在文献中已有的结论,我们给出其推广形式.

An element a in a general ring(with or without unity) is called clean if it is the sum of an idempotent and an element in Q(R), and I is called a clean general ring if each of its elements is clean. Motivated by the concepts of clean index and weakly clean index of rings,we introduce a new class of index for a general ring which is called weakly clean general index.In this article, we give some different properties about this index and show that general rings of weakly clean general index 1 are precisely those rings that are abelian rings. Furthermore,we characterize the general rings of weakly clean general indices 2 or 3. We also obtain some various properties of this kind of index for M_n(R). For some of the conclusions having been put forward in the literature, we give their extension forms.