Strong $J$-Cleanness of Formal Matrix Rings

TL;DR

This paper characterizes when elements of certain formal matrix rings over local rings are strongly J-clean, extending the understanding of ring element decompositions involving idempotents and Jacobson radicals.

Contribution

It provides necessary and sufficient conditions for strong J-cleanness in formal matrix rings over local rings, a novel extension in ring theory.

Findings

Characterization of strongly J-clean elements in M_2(R;s)

Conditions involving idempotents and Jacobson radicals

Extension of strong J-cleanness to formal matrix rings

Abstract

An element $a$ of a ring $R$ is called \emph{strongly $J$-clean} provided that there exists an idempotent $e\in R$ such that $a-e\in J(R)$ and $ae=ea$. A ring $R$ is \emph{strongly $J$-clean} in case every element in $R$ is strongly $J$-clean. In this paper, we investigate strong $J$-cleanness of $M_2(R;s)$ for a local ring $R$ and $s\in R$. We determine the conditions under which elements of $M_2(R;s)$ are strongly $J$-clean.