On a common generalization of symmetric rings and quasi duo rings

Let J(R) denote the Jacobson radical of a ring R. We call a ring R as J-symmetric if for any a, b, c is an element of R, abc = 0 implies bac is an element of J(R). It turns out that J-symmetric rings are a common generalization of left (right) quasi-duo rings and generalized weakly symmetric rings. Various properties of these rings are established and some results on exchange rings and the regularity of left SF-rings are generalized.