Pairs of domains where all intermediate domains are Noetherian

Abstract:

For Noetherian integral domains R and T with $R \subseteq T,(R,T)$ is called a Noetherian pair (NP) if every domain A, $R \subseteq A \subseteq T$, is Noetherian. When $\dim R = 1$ (Krull dimension) it is shown that the only NP’s are those given by the Krull-Akizuki Theorem. For $\dim R \geq 2$, there is another type of NP besides the finite integral extension, namely $(R,\tilde R)$ where $\tilde R = \bigcap {\{ {R_P}|{\text {rk}}\;P \geq 2\} }$. Further, for every NP (R, T) with $\dim R \geq 2$ there is an integral NP extension B of R with $T \subseteq \tilde B$. In all known examples B can be chosen to be a finite integral extension of R. For such NP’s it is shown that the NP relation is transitive. T may itself be an infinite integral extension R, though, and an example of this is given. It is unknown exactly which infinite integral extensions are NP’s.