The prime radical in a Jordan ring

Abstract

There are several definitions of radicals for general nonassociative rings given in literature, e.g. [1], [2], and [5]. The u-prime radical of Brown-McCoy which is given in [2], is similar to the prime radical in an associative ring. However, it depends on the particular chosen element u. The purpose of this paper is to give a definition for the Brown-McCoy type prime radical for Jordan rings so that the radical will be independent from the element chosen. Let J be a Jordan ring, x be an element in J; the operator U, is a mapping on J such that yU3=2x.(x.y)-x2.y for all y in J, or, equivalently, U.=2R!-R . If A, B are subsets of J, A UB is the set of all finite sums of elements of the forin aUb, where a is in A and b is in B.