PID pairs of rings and maximal non-PID subrings
type: Item
instance of: scholarly article
title: PID pairs of rings and maximal non-PID subrings
zbMATH Open document ID: 1226.13008
author: Ahmed Ayache
author: M. Ben Nasr
author: Noômen Jarboui
published in: Mathematische Zeitschrift
publication date: 2011-08-30T00:00:00Z
review text: Let \(R\subset S\) be a domain extension. \((R,S)\) is a principal ideal domain (shortly PID) pair if every ring \(T\), \(R\subset T\subset S\) is a PID. If \(R\) is not a field then it shows that the only PID pairs are those such that \(R\) is a PID. If \(R\) is a field then \((R,S)\) is a PID pair iff \(S\) is algebraic over \(R\). If \(R\) and \(S\) share a nonzero ideal, then \(R\) is a maximal non-PID subring of \(S\) iff \(S\) is a PID, \(R\) is not a PID and \(R\subset S\) is a minimal extension, that is it has no intermediate rings. Suppose that \(R\) is a Dedekind domain with infinitely many maximal ideals. Then \(R\) is a maximal non-PID subring of \(S\) iff \(R\) has a cyclic divisor class group generated by the class of any maximal ideal of \(R\).
Mathematics Subject Classification ID: 13B02
Mathematics Subject Classification ID: 13B22
Mathematics Subject Classification ID: 13C15
Mathematics Subject Classification ID: 13A15
zbMATH DE Number: 5944182
zbMATH Keywords: principal ideal domain, Dedekind domain, Prüfer domain
MaRDI profile type: Publication
full work available at URL: s00209-010-0687-4
OpenAlex ID: W2068301499
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DOI: 10.1007/S00209-010-0687-4
description: scientific article; zbMATH DE number 5944182