The completion and Krull's generalized principal ideal theorem on r-Noetherian rings

Abstract

A ring is called an r-Noetherian ring if every regular ideal is finitely generated. Let R be an r-Noetherian ring, let / be a regular ideal of R, and let (R)over-cap be the I-adic completion of R. We show that (R)over-cap is a Noetherian ring and dim(R)over-cap = sup {r-ht(M) vertical bar M is an element of Max(R) and I subset of M}. Let P be a prime ideal of R. We also prove that for any a is an element of reg(P), r-htP = ht(P/aR) + 1 and that if P is minimal over an n-generated regular ideal, then r-htP <= n.