Krull and Global Dimensions of Fully Bounded Noetherian Rings

Abstract

ABSTRACT. The main result of this paper states that the Krull dimension of a fully bounded Noetherian ring containing an uncountable central subfield is bounded above by its global dimension, provided that the latter is finite.The proof requires some results on projective dimensions and on localization (Corollary 4 and Theorem 11, respectively), which may be of independent interest. If P is a prime ideal in a Noetherian ring R, then P is contained in a unique clique, X, a subset of Spec(ü) defined below (Definition 6). Now in some circumstances, the set C(X) of elements of R regular modulo every element of X is an Ore set in R, and the localized ring Rx obtained by inverting the elements of C(X) has certain desirable properties. In this case, X is said to be classical (Definition 7). We prove in Theorem 8 that if R is a Noetherian ring of finite global dimension whose cliques are classical, then the classical Krull dimension of R is bounded above by its global dimension. Generalizing work of B. J. Mueller and A. V. Jategaonkar [16,13], we show that if R is a Noetherian fully bounded ring containing an uncountable set F of central units such that the difference of two distinct elements of F is still a unit, then all cliques in Spec(iï) are classical (Theorem 11). This applies, in particular, if R has an uncountable central subfield, as in the abstract. Using Theorems 8 and 11 in their general forms, K. R. Goodearl and L. W. Small have shown that the inequality of Krull and global dimensions is true for all Noetherian P. I. rings [10].In this paper, all modules are right modules unless it is indicated otherwise. If M is a module, then r(M) is the right annihilator of M and (if appropriate) l(M) is the left annihilator of M. A ring is Noetherian if it satisfies the ascending chain condition on right and left ideals. If P is a prime ideal of R, then R/P is right bounded if every essential right ideal of R/P contains a nonzero two-sided ideal. A ring R is fully bounded Noetherian (abbreviated FBN) if it is Noetherian and for every prime ideal P, R/P is both right and left bounded.Part of this research was done while the first author held a visiting position at the University of Washington. He is grateful to that institution for its hospitality. The research of the second author was supported in part by a grant from the NSF.

Smooth polynomial identity algebras with almost factorial centers

“…If (R[t] S ) X is hereditary then (P S ) X is obviously projective. If gl.dim (R[t] S ) X = 2, then since height((P S ) X ) = 1, we get by [6,Corollary 5], that (P S ) X is projective. This implies (using the fact that a module M R is projective iff 1 ∈ MM * ) that P S is a projective R[t] S -module (from both sides).…”