Links between prime ideals in differential operator rings

Abstract

In recent years much work has been done on a localization theory for noetherian rings. Links between prime ideals play a central role in this development. When a ring R is not commutative, it is rarely possible to localize R at a prime ideal P. Having a link between P and another prime ideal is an obstruction to localization. In this paper we will investigate the links between prime ideals in differential operator rings, showing that it is possible to localize these rings in a certain sense. Recall that a multiplicatively closed subset S of a ring R is a right Ore set if for all r in R and all s in S we have rS n sR # Qr. The set S is an Ore set if it is both a right and a left Ore set. It is always possible to construct a ring RF’ where the elements of S become units. However, if we insist on writing the elements of RF ’ as right fractions, rs ’ with r in R and s in S, then the set S must be a right Ore set [9, Sect. 12.11. Let P be a prime ideal of R, and let %7(P) = {r E R 1 r + P is regular in R/P). We say that P is a localizable prime ideal if the set U(P) is an Ore set. We will shortly give an example of a prime ideal which is not localizable, but first we introduce the differential operator ring. Let R be a ring, and let 6 be an additive map from R to itself. We call 6 a derivation if it satisfies the product rule; 6(ab)=@a) b+ad(b). The differential operatqr ring over R, denoted R[0; S], is a free left R-module with basis 1, 8, 13~ ,.... Thus every element of R[0; S] is a polynomial in 0 with coefficients from R. The addition in R[l?; S] is defined as usual for polynomials, but multiplication is extended from R by the rule Br = rtI + 6(r). The differential operator ring is an associative ring and is determined up to isomorphism by an obvious universal property [9, Sect. 12.21.