On classical quotients of polynomial identity rings with involution

Abstract:

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