Algebraic conditions on rings with involution

Abstract

Herstein, Procesi, and Schacher have recently shown [ 13, Corollary3.71, that if R is a division ring with center % satisfying (xy -~ ys)“‘(.“.“) E Z:, n(.v, ~1) a positive integer depending on s and y, then dim,R < 4. Here we generalize this result to rings with involution. If .T, ,..., x,, are elements of a ring R, we use the notation [.q ,..., sII] for [[sr , .x,,], s:r] ,... 1, s,)] where [x, y] .vy -~ye. In Theorem 5 we shoMthat if R is a primitive ring with involution -i satisfying