The Levitzki radical in associative and Jordan rings

Abstract

If R is an associative ring in which 2x = a has a unique solution for all a E R, then it is of interest to consider the attached ring R-, where R+ is the same additive group as R but multiplication in Rf is given by a . 6 = $(ab + ba) (Here ab represents the multiplication in R and $a is the element x for which 2x = a). R+ is a linear Jordan ring by virtue of the fact that it satisfies the identity x2 . (x . u) = R . (x” .J) f an 1 s ine rizations. Similarly, d ‘t 1 a if R is equipped with an involution *, then we also have the attached subring S of R+ where S is the set of *-symmetric elements of R. There are many known interrelationships between R, R+, and S. For example, Herstein [6, 71 has shown that R is simple if and only if R+ is simple if and only if S is simple. NIcCrimmon [9] extended this result to arbitrary rings not necessarily satisfying the condition that 2~ = a has a solution. In addition Rad R = Rad R+, where Rad denotes the Jacobson, nil, or prime radical [3, 91. Finally, Rad S = S n Rad R if Rad denotes the Jacobson or prime radical [3,9]. In this paper, we compare Z(R), B(R+), and T(S) where Y denotes the Levitzki radical. Unless otherwise stated (Lemma 5 and Theorem 2) we will assume that R satisfies the characteristic condition mentioned above. In Section 2 we show that Z(R) = Z(R-t) and as a by-product we see that there exist finitely generated nil Jordan algebras that are not nilpotent. In Section 3, we study the relationship between 5?(S) and S n X(R) and show that P(S) = S n 8(R). Along the way we show that if R is an algebra over a field F with “enough” elements, then if S is nil of index n, then R is nil of index < 2n.