Noetherian intersections of integral domains

Abstract:

Let $D < R$ be integral domains having the same quotient field K and suppose that there exists a family ${\{ {V_i}\} _{i \in I}}$ of 1-dim quasi-local domains having quotient field K such that $D = R \cap \{ {V_i}|i \in I\}$. The goal of this paper is to find conditions on R and the ${V_i}$ in order for D to be noetherian and, conversely, conditions on D in order for R and the ${V_i}$ to be noetherian. An important motivating case is when the set $\{ {V_i}\}$ consists of a single element V and V is a valuation ring. It is shown, for example, in this case that (i) if V is centered on a finitely generated ideal of D, then V is noetherian and (ii) if V is centered on a maximal ideal of D, then D is noetherian if and only if R and V are noetherian.