n-Clean Rings
n-Clean Rings
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DOI：
10.1142/s1005386706000538
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发表时间：
2006-12
影响因子：
0.3
通讯作者：
Guangshi Xiao;W. Tong
中科院分区：
文献类型：
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作者：
Guangshi Xiao;W. Tong
设n为正整数。一个环R称为n-clean，如果R的每个元素都可以表示为R中的一个幂等元与n个单位元之和。n-clean环类包含clean环和（S，n）-环（即，每个元素是不超过n个单位的和）。本文研究了n-clean环的一些性质。存在一个不是（S，2）-环的clean and（S，3）-环.若R是满足（SI）的环，则多项式环R[x]对任何正整数n都不是n-clean的。通过一个例子证明了对任意正整数n> 1，存在一个非n-clean环R，使得R上的2× 2矩阵环M2（R）是n-clean的.
Let n be a positive integer. A ring R is called n-clean if every element of R can be written as a sum of an idempotent and n units in R. The class of n-clean rings contains clean rings and (S,n)-rings (i.e., every element is a sum of no more than n units). In this paper, we investigate some properties on n-clean rings. There exists a clean and (S,3)-ring which is not an (S,2)-ring. If R is a ring satisfying (SI), then the polynomial ring R[x] is not n-clean for any positive integer n. An example shows that for any positive integer n> 1, there exists a non n-clean ring R such that the 2× 2 matrix ring M2(R) over R is n-clean.