On feckly clean rings

TL;DR

This paper characterizes feckly clean rings through various equivalent conditions involving idempotents, prime ideals, and the Jacobson radical, providing a comprehensive understanding of their structure.

Contribution

It offers new equivalent conditions and characterizations for feckly clean rings, including topological and ideal-theoretic perspectives.

Findings

Feckly clean rings are characterized by the existence of specific idempotents related to prime ideals.

The Jacobson radical plays a key role in the structure of feckly clean rings.

Strong zero-dimensionality of certain spectra characterizes feckly cleanness.

Abstract

A ring $R$ is feckly clean provided that for any $a\in R$ there exists an element $e\in R$ and a full element $u\in R$ such that $a=e+u, eR(1-e)\subseteq J(R)$. We prove that a ring $R$ is feckly clean if and only if for any $a\in R$, there exists an element $e\in R$ such that $V(a)\subseteq V(e), V(1-a)\subseteq V(1-e)$ and $eR(1-e)\subseteq J(R)$, if and only if for any distinct maximal ideals $M$ and $N$, there exists an element $e\in R$ such that $e\in M, 1-e\in N$ and $eR(1-e)\subseteq J(R)$, if and only if $J$-$spec(R)$ is strongly zero dimensional, if and only if $Max(R)$ is strongly zero dimensional and every prime ideal containing $J(R)$ is contained in a unique maximal ideal. More explicit characterizations are also discussed for commutative feckly clean rings.