Strongly Divided Pairs of Integral Domains

Abstract

This work generalizes the recent study of the class of strongly divided (commutative integral) domains. Let $R \subseteq T$ be domains with (R, m) quasi-local. Then (R, T) is said to be a strongly divided pair if, for each ring E such that $R \subseteq E \subseteq T$ and each $Q \in \mathrm {Spec}(E)$ such that $Q\cap R \subset m$, one has $Q \subset R$. Let $\overline{R}$ be the integral closure of R in T. Then (R, T) is a strongly divided pair if and only if R and $\overline{R}$ have the same sets of nonmaximal prime ideals and, for each maximal ideal M of $\overline{R}$, $(\overline{R}_M, T_M)$ is a strongly divided pair. If R is integrally closed in T and R is treed, then (R, T) is a strongly divided pair if and only if R[u] is a treed domain for each $u \in T$. If $mT=T$ and R is integrally closed in T, then (R, T) is a strongly divided pair if and only if $T=R_p$ for some divided prime ideal p of R and R / p is a strongly divided domain. Examples of strongly divided pairs ((R, m), T) such that $mT \ne T$ are given using pullbacks with data having prime spectra pinched at some nonmaximal prime ideal. Additional results and examples are given to illustrate the theory and its sharpness.