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

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