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### The Levitzki Radical in Jordan Rings

Chester E. Tsai
1970 Proceedings of the American Mathematical Society
The main purpose of this paper is to give an external characterization of the Levitzki radical of a Jordan ring 31 as the intersection of a family of prime ideals 21. This characterization coincides with that of associative rings which was given by Babic in [l]. 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
more » ... in associative rings is called the Baer radical. If the minimal condition on ideals holds on Jordan ring 21, then the Levitzki radical, P(2l), and the prime radical, P(2l) of 21 coincide. Throughout this paper, any Jordan ring 21, that is a (nonassociative) ring satisfying (1) ab = ba, and (2) a2iab) =aia2b) for all a, b in 21, and any of its subrings satisfy the conditions, 2a = 0 implies a = 0 and (4) if a is in a subring C of 21 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 . (*) R(.xy)l = RxRyz + R"Rzx + RZRXy -RxRzRy ~ RVRZRX where, for any element u in 21, R" is the formal multiplication of u in the ring 21, i.e., aRu = au for all a in 21. If A, B are subsets of 21, A Ub denotes the set of finite sums of elements of the form a Ub where a is in A and b is in B and aUb = 2 iab)b -ab2. If A and P are ideals in 2T, then A3, iAB)B+AiBB) and A UB are ideals in 21. The first two are proved in  . To prove A Ub is an ideal, let a be an element in A and b, c be elements in P and aU(b.C) = \aiUb+c-Ub-Uc) = iab)c+iac)b -ibc)a. Assume that both A and B are ideals in 21, and applying (*) one can easily check that for any u in 21, iaU(b,c))-u=aU(b',C)+aU(b,c>)-a'U(b,c) where a' = au, b' = bu, and c' = cu. Letting c -b one sees that A Ub is an ideal in 21 if A and P are. It is proved in  that if A is an ideal in 21, then A3 = A Ua-If 21 is a Jordan ring, we denote 2T3 by 2fi, and, inductively 2li+i = 21? for each i. We also denote 2l(o> as 21 and, inductively 2l(t+u =2l(i)2I«)