A generalization of the prime radical in nonassociative rings

Abstract

In [5] Tsai defined the Brown-McCoy prime radical for Jordan rings in terms of the quadratic operation and proved basic results for the radical. In this paper we give a definition of the prime radical for arbitrary nonassociative rings in terms of a ^-operation defined on the family of ideals and of a function / of the ring into the family of ideals in the ring. The prime radical for Jordan or standard rings is obtained by a particular choice of the ^-operation and the function /. We also extend the results for the Jordan case to weakly W admissible rings which include the generalized standard rings and therefore alternative and standard rings as well as Jordan rings.