Chain conditions on symmetric elements

Abstract

Recently Britten has proven an analog of Goldie’s theorem for the Jordan ring S S of symmetric elements in a ring with involution of characteristic not 2. In this paper we first extend Britten’s theorem to the situation where R R is an arbitrary ring and the Jordan ring is only an ample subring of the symmetric elements. We apply this result to show that if S S has ACC on quadratic ideals, then the (Jordan) nil radical of S S is nilpotent.