The Levitzki radical in Jordan rings

Abstract

The main purpose of this paper is to give an external characterization of the Levitzki radical of a Jordan ring 2f as the intersection of a family of prime ideals W. This characterization coincides with that of associative rings which was given by Babic in [1I]. Applying this characterization, it is easy to see that the Levitzki radical of a Jordan ring contains the prime radical of the same ring. For associative rings the same statement is well known, since the prime radical in associative rings is called the Baer radical. If the minimal condition on ideals holds on Jordan ring 2, then the Levitzki radical, L(2f), and the prime radical, R(2f) of 2f coincide. Throughout this paper, any Jordan ring 2f, that is a (nonassociative) ring satisfying (1) ab = ba, and (2) a2(ab) =a(a2b) for all a, b in 2t, and any of its subrings satisfy the conditions, (3) 2a = 0 implies a = 0 and (4) if a is in a subring C of 2 then there exists a unique element x in C such that 2x = a. In a Jordan ring, the following identity (*) is well known. One can find the proof in [3 ].