On classical quotients of polynomial identity rings with involution

Let (R, *) denote a ring R with involution (*), where “involution„ means “anti-automorphism of order ≦ two„. We can specialize many ring-theoretical concepts to rings with involution; in particular an ideal of (R, *) is an ideal of R stable under (*), and the center of (R, *) is the set of central elements of R which are fixed under (*). Then we say (R, *) is prime when the product of any two nonzero ideals of (R, *) is nonzero; similarly (R, *) is semiprime when any power of a nonzero ideal of (R, *) is nonzero. The main result of this paper is a strong analogue to Posner’s theorem [5], namely that any prime (R, *) with polynomial identity has a ring of quotients RT, formed merely by adjoining inverses of nonzero elements of the center of (R, *). This quotient ring (RT, *) is simple and finite dimensional over its center. An extension of these results to semiprime Goldie rings with polynomial identity is given.