The prime radical in special Jordan rings

Abstract:

If R is an associative ring, we consider the special Jordan ring ${R^ + }$, and when R has an involution, the special Jordan ring S of symmetric elements. We first show that the prime radical of R equals the prime radical of ${R^ + }$, and that the prime radical of R intersected with S is the prime radical of S. Next we give an elementary characterization, in terms of the associative structure of R, of primeness of S. Finally, we show that a prime ideal of R intersected with S is a prime Jordan ideal of S.