A Note on Clean Rings

• 来源期刊: 代数集刊(英文版)
• 关键词: clean rings 9(x)-clean rings group rings
• 年，卷（期）: 2007,（3）
• 页码范围: 537-540
• 页数: 4页
• 语种: 英文

摘要

Let R be a ring and g(x) a polynomial in C[x],where C=C(R) denotes the center of R.Camillo and Sim6n called the ring g(x)-clean if every element of R can be written as the sum of a unit and a root of g(x).In this paper,we prove that for a,b (E) C,the ring R is clean and b - a is invertible in R if and only if R is g1(x)-clean,where gl(x) = (x - a)(x - b).This implies that in some sense the notion of g(x)-clean rings in the Nicholson-Zhou Theorem and in the Camillo-Sim6n Theorem is indeed equivalent to the notion of clean rings.