Publications de l’Institut Mathématique, Nouvelle Série Vol. 102[116], pp. 133–148 (2017) 

Rings in which the power of every element is the sum of an idempotent and a unitHuanyin Chen, Marjan SheibaniDepartment of Mathematics; Hangzhou Normal University, Hangzhou, China; Faculty of Mathematics, Statistics and Computer Science, Semnan University, Semnan, IranAbstract: A ring $R$ is uniquely $\pi $clean if the power of every element can be uniquely written as the sum of an idempotent and a unit. We prove that a ring $R$ is uniquely $\pi $clean if and only if for any $a\in R$, there exists an integer $m$ and a central idempotent $e\in R$ such that ${a}^{m}e\in J\left(R\right)$, if and only if $R$ is Abelian; idempotents lift modulo $J\left(R\right)$; and $R/P$ is torsion for all prime ideals $P\supseteq J\left(R\right)$. Finally, we completely determine when a uniquely $\pi $clean ring has nil Jacobson radical. Keywords: idempotent unit; Jacobson radical; uniquely clean ring; $\pi $uniquely clean rings Classification (MSC2000): 16S34; 16U60; 16U99; 16E50 Full text of the article: (for faster download, first choose a mirror)
Electronic fulltext finalized on: 3 Nov 2017. This page was last modified: 29 Jan 2018.
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