On uniquely $\pi$-clean rings

TL;DR

This paper characterizes uniquely  rings, showing they are abelian with specific lifting and torsion properties, and explores conditions for such rings with nil Jacobson radical.

Contribution

It provides a complete characterization of uniquely  rings, linking algebraic properties to idempotent and radical conditions.

Findings

A ring is uniquely  iff it is abelian, with lifted idempotents and torsion quotient.

Such rings with nil Jacobson radical are abelian periodic rings.

Existence of a unique idempotent for powers of elements relates to prime radical inclusion.

Abstract

An element of a ring is unique clean if it can be uniquely written as the sum of an idempotent and a unit. A ring $R$ is uniquely $\pi$-clean if some power of every element in $R$ is uniquely clean. In this article, we prove that a ring $R$ is uniquely $\pi$-clean if and only if for any $a\in R$, there exists an $m\in {\Bbb N}$ and a central idempotent $e\in R$ such that $a^m-e\in J(R)$, if and only if $R$ is abelian; every idempotent lifts modulo $J(R)$; and $R/P$ is torsion for all prime ideals $P$ containing the Jacobson radical $J(R)$. Further, we prove that a ring $R$ is uniquely $\pi$-clean and $J(R)$ is nil if and only if $R$ is an abelian periodic ring, if and only if for any $a\in R$, there exists some $m\in {\Bbb N}$ and a unique idempotent $e\in R$ such that $a^m-e\in P(R)$, where $P(R)$ is the prime radical of $R$.