The strong $P$-cleanness over rings

TL;DR

This paper characterizes rings where every element can be expressed as the sum of an idempotent and a strongly nilpotent element, providing criteria and characterizations, including for matrices over local rings.

Contribution

It offers necessary and sufficient conditions for rings to be strongly $P$-clean and characterizes such rings, including matrix rings over local rings.

Findings

Characterization of strongly $P$-clean rings

Criteria for $2\times 2$ matrices over local rings

Conditions for strong $P$-cleanness in rings

Abstract

An element of a ring $R$ is strongly $P$-clean provided that it can be written as the sum of an idempotent and a strongly nilpotent element that commute. A ring $R$ is strongly $P$-clean in case each of its elements is strongly $P$-clean. We investigate, in this article, the necessary and sufficient conditions under which a ring $R$ is strongly $P$-clean. Many characterizations of such rings are obtained. The criteria on strong $P$-cleanness of $2\times 2$ matrices over commutative local rings are also determined.