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Strongly Divided Pairs of Integral Domains

  • Ahmed Ayache
  • David E. DobbsEmail author
Chapter
Part of the Trends in Mathematics book series (TM)

Abstract

This work generalizes the recent study of the class of strongly divided (commutative integral) domains. Let \(R \subseteq T\) be domains with (Rm) quasi-local. Then (RT) 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 (RT) 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 (RT) 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 (RT) 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 ((Rm), 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.

Keywords

Integral domain Overring Prime ideal Treed domain Pullback Integrality Pseudo-valuation domain Strongly divided domain Krull dimension Field 

2010 Mathematics Subject Classification

Primary: 13G05 13B99 Secondary: 13A15 13B21 

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Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Faculty of Science, Department of MathematicsUniversity of BahrainSukhirBahrain
  2. 2.Department of MathematicsUniversity of TennesseeKnoxvilleUSA

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