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 is an element of R, ab = 0 implies ba is an element of 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.