J-Clean and Strongly J-Clean Rings (Communications in Mathematical Research, 2019)
Let $R$ be a ring and $J(R)$ the Jacobson radical. An element $a$ of $R$ iscalled (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 providedthat 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 resultsare also extended.
Keywords: J-clean ring; strongly J-clean ring; generalized matrix ring