On rings with central polynomials

Abstract

Let Ω be a commutative ring with 1, and let R be an Ω-algebra with center C. A polynomial f( X 1,…, X m ) with coefficients in Ω is an identity of R if f( r 1,…, r m ) = 0 for all r 1,…, r m in R, and f is central in R if f is not an identity of R but f( r 1,…, r m ) ϵ C for all r 1,…, r m in R. Formanek has shown recently that M n(Ω) , the algebra of n × n matrices with entries in Ω, has a central polynomial which is linear in the last variable. Call such a central polynomial regular. It follows from Formanek's result that a large class of algebras have regular central polynomials. The purpose of this paper is to examine ideals of rings with regular central polynomial and to see what relationships exist between these ideals and ideals of the center. In order to do this it is necessary to develop some general results on identities of algebras and to investigate central localization, i.e., localization of R by a multiplicative subset S of C. As an application of the results of Sections 1–4, we obtain some information on semiprime P.I.-algebras in Section 5 and conclude with a fairly straight-forward proof in Section 6 of the Artin-Procesi theorem, which says that a ring R (with 1) is Azumaya of rank n 2 over its center if and only if R satisfies the identities of M n ( Z ) and no homomorphic image of R satisfies the identities of M n−1 ( Z ).