SYMMETRIC PROPERTY OF RINGS WITH RESPECT TO THE JACOBSON RADICAL

Let R be a ring with identity and J(R) denote the Jacobson radical of R, i.e., the intersection of all maximal left ideals of R. A ring R is called J-symmetric if for any a, b, c is an element of R, abc = 0 implies bac is an element of J(R). We prove that some results of symmetric rings can be extended to the J-symmetric rings for this general setting. We give many characterizations of such rings. We show that the class of J-symmetric rings lies strictly between the class of symmetric rings and the class of directly finite rings.