Intermediary rings in normal pairs
type: Item
instance of: scholarly article
title: Intermediary rings in normal pairs
zbMATH Open document ID: 1148.13003
author: Ahmed Ayache
author: Noômen Jarboui
published in: Journal of Pure and Applied Algebra
publication date: 2008-07-10T00:00:00Z
review text: All rings considered are integral domains. A pair of rings \((R,S)\) is called a \textit{normal pair} if \(R\subseteq S\) and every ring contained between \(R\) and \(S\) is integrally closed in \(S\). Given a ring extension \(R\subset S\), the symbol \([R,S]\) denotes the set of all rings between \(R\) and \(S\), including the end-points. The authors are interested in the cardinality of the latter interval of rings. Notation needed for the main result is as follows: \(\text{Supp}(S/R)=\{ P\in \text{Spec}(R) \mid PS=S\}\) (as an ordered set) and it is assumed to contain \(n\) maximal elements. For a non-zero ideal \(I\) of \(R\), denote \(\Omega_R(I)=\{ x\in qf(R) \mid \forall y\in I\exists n\geq 1\,\, xy^n\in R\}\) where qf stands for the quotient field; define inductively \(\Gamma_1=\{\{Q\} \mid Q\in \text{Supp}(S/R)\}\) and for \(2\leq k\leq n\) denote by \(\Gamma_k\) the set of all subsets \(\{Q_1,\dots,Q_k\}\) of \(\text{Supp}(S/R)\) made up of \(k\) pairwise incompatible elements. The main results is: Theorem 2.4. Every \(T\in [R,S]\setminus \{R\}\) is uniquely represented as \(T=\Omega_R(Q_1Q_2\dots Q_k)\), where \(\{Q_1,Q_2,\dots Q_k\}\in \bigcup_{i=1}^n\Gamma_i\). Consequently, the cardinality of \([R,S]\) is \(1+| \text{Supp}(S/R)| + \sum_{i=2}^n| \Gamma_i| \). Corollary 2.5. \(| \text{Supp}(S/R)| +2^n-n\leq | [R,S]| \leq 2^{| \text{Supp}(S/R)| }\). Corollary 2.7. \([R,S]\) is totally ordered iff its cardinality is by 1 greater than the cardinality of \(\text{Supp}(S/R)\), iff \(R\) has a unique maximal overring in \([R,S]\), iff \(\text{Supp}(S/R)\) is totally ordered. Theorem 3.1 \(\text{Supp}(S/R)\) is contained in the set of maximal ideals of \(R\) iff the cardinality of \([R,S]\) is \(2^{| \text{Supp}(S/R)| }\). Corollary 3.2. If \(R\) is noetherian, then the cardinality of \([R,S]\) is \(2^m\), where \(m\) is the set of maximal ideals \(M\) of \(R\) with \(\text{ht}(M)=1\) and \(MS=S\).
reviewed by: Radoslav M. Dimitrić
Mathematics Subject Classification ID: 13B02
Mathematics Subject Classification ID: 13C15
Mathematics Subject Classification ID: 13A18
Mathematics Subject Classification ID: 13B25
Mathematics Subject Classification ID: 13E05
zbMATH DE Number: 5299435
zbMATH Keywords: overring
zbMATH Keywords: normal pair of rings
zbMATH Keywords: intermediate rings
zbMATH Keywords: integral closure
zbMATH Keywords: Kaplansky transform of a ring with respect to an ideal
zbMATH Keywords: height of an ideal
zbMATH Keywords: prime spectrum
zbMATH Keywords: maximal spectrum
zbMATH Keywords: Supp(S/R)
zbMATH Keywords: totally ordered set of rings
zbMATH Keywords: Noetherian domain
zbMATH Keywords: Prüfer domain
MaRDI profile type: Publication
full work available at URL: j.jpaa.2008.03.002
OpenAlex ID: W2074572339
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DOI: 10.1016/J.JPAA.2008.03.002
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description: scientific article; zbMATH DE number 5299435