The Prime Radical in a Jordan Ring

Chester Tsai
1968 Proceedings of the American Mathematical Society  
There are several definitions of radicals for general nonassociative rings given in literature, e.g. [l], [2] , and [5] . The M-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 / be a Jordan ring, x be an
more » ... n ring, x be an element in J; the operator Ux is a mapping on J such that yUx -2x-(x-y)-x2-y for all y in J, or, equivalently, UX = 2R\ -R\. If A, B are subsets of J, A UB is the set of all finite sums of elements of the form aUt, where a is in A and ft is in B. Lemma 1. Let P be a two sided ideal in J. Then the following three statements are equivalent.
doi:10.2307/2036054 fatcat:rq5hfuqsyzautjaczaojc65sha