J-Clean and Strongly J-Clean Rings

Abstract

Let $R$ be a ring and $J(R)$ the Jacobson radical. An element $a$ of $R$ is called (strongly) $J$-clean if there is an idempotent $e\in R$ and $w\in J(R)$ such that $a=e+w$ (and $ew=we$). The ring $R$ is called a (strongly) $J$-clean ring provided that every one of its elements is (strongly) $J$-clean. We discuss, in the present paper, some properties of $J$-clean rings and strongly $J$-clean rings. Moreover, we investigate $J$-cleanness and strongly $J$-cleanness of generalized matrix rings. Some known results are also extended.