Reversibility of Rings with Respect to the Jacobson Radical

Let R be a ring with identity and J ( R ) denote the Jacobson radical of R . A ring R is called J - reversible if for any a , b ∈ R , a b = 0 implies b a ∈ J ( R ) . In this paper, we give some properties of J -reversible rings. We prove that some results of reversible rings can be extended to J -reversible rings for this general setting. We show that J -quasipolar rings, local rings, semicommutative rings, central reversible rings and weakly reversible rings are J -reversible. As an application it is shown that every J -clean ring is directly finite.