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.