Localization in ore extensions of commutative noetherian rings

Abstract

There is extensive literature on the subject of localization at prime ideals in Noetherian rings. In the non-commutative setting, where it is seldom possible to localize at a prime ideal, one central concept which has emerged is the notion of a “link” between prime ideals as an obstruction to localization. In recent years, much work has been devoted to describing linked prime ideals in several important classes of non-commutative Noetherian rings, notably group rings (K. A. Brown in [4, 8]), universal enveloping algebras (K. A. Brown in [S, 6]), and differential operator rings (G. Sigurdsson in [ 173). In this paper, we analyze the localizability of prime ideals in an Ore extension of a ring R. If R is a ring with automorphism g, the Ore extension (or skew polynomial ring) R[x; a] is defined to be the set of polynomials of the form Cy=, r,xi where riE R. With termwise addition and with multiplication determined by distributivity and the relation xr = r”x, R[x; a] is an associative ring. If R is (right) Noetherian, the Hilbert Basis Theorem shows that R[x; a] is also (right) Noetherian. In this paper, we shall assume that the coefficient ring R is a commutative Noetherian ring. All rings are associative with an identity element. A ring is Noetherian if it is both right and left Noetherian; the same convention applies to other ambidextrous properties. For an R-bimodule M, I(M), and r(M) are, respectively, the left and right annihilators of M in R. For an ideal A of R, V,JA) denotes the collection of elements of R which are regular modulo A. Earlier versions of the results in this paper appeared in the author’s Ph.D. thesis where Krull dimension techniques were used extensively. The author would like to thank his thesis advisor at McMaster University,