Prime rings with involution and generalized polynomial identities

Abstract

Our main purpose in writing this paper is to prove that if R is a prime ring with involution whose symmetric elements satisfy a generalized polynomial identity over the extended centroid C of R, then the central closure A = RC + C must in fact be a primitive ring with a minimal right ideal eA such that eAe is a finite dimensional division algebra over C. This generalizes a previous theorem of ours [8], in which we obtained the above result for the case where R was assumed to be primitive. It also generalizes a recent result of Skinner [lo, Theorem 5.11, where the above result was obtained for the case where R was a prime Goldie ring. In the course of proving this theorem, and without digressing too much from our main goal, we rework and combine the techniques of Amitsur in [l] and [2] with those of ours in [7], so as to obtain the various structure theorems (due to Amitsur, Posner, Herstein, Kaplansky, and the author) on simple, primitive, and prime rings (with involution) whose (symmetric) elements satisfy a (generalized) polynomial identity. We hope this attempt to give a more or less unified approach to a group of theorems, the existing proofs of which are not for the most part too closely related, will be of general interest. In Section 2, we recall the notion of extended centroid of a prime ring and discuss some of its key properties (Theorems 2.1-2.4). Next, putting together some ideas of Amitsur, we give a particular way of embedding prime rings in primitive rings (Theorems 2.5-2.8). Finally, we show how information about primitive rings can be pulled back to prime rings (Theorems 2.9-2.10). In Section 3, we recall the notion of generalized multilinear identity (GMI) and show (Theorems 3.1-3.2) that GMI’s are carried over by the aforementioned embedding of prime rings in primitive rings. A fundamental result of Amitsur (Theorem 3.3) together with its corollary, Theorem 3.5, are applicable to primitive rings and say in effect that some nonzero linear