(untitled) (Communications in Mathematical Research, 2019)
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 eR and wJ(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.