Nil-clean and strongly nil-clean rings

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• 作者：Tamer Ko艧an^a ; Zhou Wang^b ; Yiqiang Zhou^c ; ^{zhou@mun.ca" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Primary ; 16U99 ; secondary ; 16E50 ; 16N40 ; 16S34 ; 16S50 ; 16S70
• 刊名：Journal of Pure and Applied Algebra
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：220
• 期：2
• 页码：633-646
• 全文大小：405 K

文摘

An element a of a ring R   is nil-clean if a=e+b$a=e+b$ where e²=e∈R$e^2=e\in R$ and b   is a nilpotent; if further eb=be$eb=be$, the element a is called strongly nil-clean. The ring R is called nil-clean (resp., strongly nil-clean) if each of its elements is nil-clean (resp., strongly nil-clean). It is proved that an element a is strongly nil-clean iff a   is a sum of an idempotent and a unit that commute and a−a²$a-a^2$ is a nilpotent, and that a ring R   is strongly nil-clean iff R/J(R)$R/J(R)$ is boolean and J(R)$J(R)$ is nil, where J(R)$J(R)$ denotes the Jacobson radical of R. The strong nil-cleanness of Morita contexts, formal matrix rings and group rings is discussed in details. A necessary and sufficient condition is obtained for an ideal I of R   to have the property that R/I$R/I$ strongly nil-clean implies R is strongly nil-clean. Finally, responding to the question of when a matrix ring is nil-clean, we prove that the matrix ring over a 2-primal ring R   is nil-clean iff R/J(R)$R/J(R)$ is boolean and J(R)$J(R)$ is nil, i.e., R is strongly nil-clean.