一般Clean环的扩张

Extension of Clean General Rings

有单位元的结合环R称为clean环,如果R中的每个元素都可以写成一个幂等元与一个单位的和的形式.W.K.Nicholson,Y.Zhou[1]给出了一般clean环的定义.在此基础上讨论了一般clean环的几个扩张性质,得到如下结论:1)一般环I是一般clean环当且仅当I的形式幂级数环是一般clean环当且仅当I的斜幂级数环是一般clean环.2)若I是一般clean环,则对任意n≥1,均有I[x]/<xn+1>是一般clean环.

An associative ring with unity is called clean if every element is the sum of an idempotent and a unit. W. K. Nicholson, Y. Zhou defined clean general rings. Some extension of clean general rings is expounded based on it and it is shown that： （1） a general ring I is a clean general ring if and only if its power series ring is a clean general ring if and only if its skew power series ring is a clean general ring. （2） If I is a clean general ring, then I[x]〈x^n＋1〉 is a clean general ring for all n≥1.