RINGS IN WHICH THE POWER OF EVERY ELEMENT IS THE SUM OF AN IDEMPOTENT AND A UNIT


Huanyin Chen, Marjan Sheibani




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 $a^m-e\in J(R)$, if and only if $R$ is Abelian; idempotents lift modulo $J(R)$; and $R/P$ is torsion for all prime ideals $P\supseteq J(R)$. Finally, we completely determine when a uniquely $\pi$-clean ring has nil Jacobson radical.