Rings in which the power of every element is the sum of an idempotent and a unit

Huanyin Chen, Marjan Sheibani

Abstract: 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

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