Noetherian extensions of commutative rings

Abstract

Let A⊆B be an extension of commutative rings with identity and T(A) (resp., T(B)) be the total quotient ring of A (resp., B). We say that A is a B-Noetherian ring if every ideal I of A with IB=B is finitely generated, and the extension A⊆B is called a Noetherian extension if A is B-Noetherian. In this paper, among other things, we introduce the ring Ag(B), the global transform of A in B. We then show that if x∈A is B-regular (i.e., xB=B) and R is a ring between A and Ag(B), then R/xR is a Noetherian A-module and every ideal of R containing x is finitely generated. We also study some conditions on A and B under which if A is B-Noetherian with dimBA=1, then any ring between A and B is B-Noetherian. This is the Krull-Akizuki theorem when A is an integral domain and B=T(A). Finally, we prove that if T(B) is integral over T(A) and A is integrally closed in B, then every ideal I of A with IB=B is t-invertible, i.e., (II−1)t=A.