The Prime Radical in Alternative Rings

The characterization by J. Levitzki of the prime radical of an associative ring R as the set of strongly nilpotent elements of R is adapted here to apply to a wide class of nonassociative rings. As a consequence it is shown that the prime radical is a hereditary radical for the class of alternative rings and that the prime radical of an alternative ring coincides with the prime radical of its attached Jordan ring. In 1951 J. Levitzki characterized the prime radical of an associative ring R as the set of all elements r G R such that every m-sequence beginning with r ends in zero [3]. An m-sequence was defined to be a sequence (a0,a,,..., an,...} such that a¡ G íj,_, Äa,_, for i = 1, 2, .... Recently, C. Tsai [9] has given a similar characterization for the prime radical of a Jordan ring. Here we extend this characterization to the class of all i-rings and, as a consequence, are able to show that the prime radical is a hereditary radical on the class of alternative rings (i.e., if A is an ideal of an alternative ring R, then P(A) = A n P(R)) and that P(R) = P(R+) for all 2 and 3-torsion free alternative rings R. Although it is not known whether the prime radical is hereditary on the class of all i-rings, a partial result in this direction is obtained. Recall that a not necessarily associative ring R is called an i-ring for a positive integer s if As is an ideal of R whenever A is an ideal of R (As denotes the set of all sums of products a, a2 • • • as for a¡ G A under all possible associations). An ideal P of R is called a prime ideal if whenever AXA2 • • ■ As S P then A j Q P for some i. Here AXA2 • • • As denotes the product of the ideals under all possible associations. The prime radical, P(R), of R is the intersection of all prime ideals of R and can be characterized as the set of all elements r G R such that every complementary system M of R which contains r also contains 0. A set M in R is a complementary system if whenever A{, A2, ..., As are s ideals of R such that A¡ D M ¥= 0 for /' = 1,2,..., s, then (AlA2---As) n M # 0 [6], [8], [10]. To make this article self-contained we mention the following three properties of P(R) which hold for any í-ring R. Proofs can be found in [6], [8] and [10]. (a) P(R) = 0 if and only if R contains no nonzero nilpotent ideals. (b) P(R/P(R)) = 0. Received by the editors May 29, 1975. AMS (MOS) subject classifications (1970). Primary 17D05, 17E05.