A new radical called thej-radical is defined forS, the symmetric elements, determined by the Jordan multiplication rather than on the quadratic multiplication for prime radicals. If the ring is 2-torsion free then these two notions are seen to be equivalent. Next some results dealing with the prime and Levitzki radicals of the ring and the subring [Sbar], generated by the symmetric elements, are proved. Whenever 2R = Rthen [Sbar] inherits the prime and Levitzki radical of the ring.