The dimension-overrings equation and maximal ideals of integral domains
type: Item
instance of: scholarly article
title: The dimension-overrings equation and maximal ideals of integral domains
zbMATH Open document ID: 1542.13006
author: Ali Jaballah
published in: Beiträge zur Algebra und Geometrie
publication date: 2024-05-31T00:00:00Z
review text: Let \(R\) be an integral domain with quotient field \(K\) and \(O(R)\) the set of all overrings of \(R\). In this paper, the author investigated integral domains with only finitely many overrings and establish several new sharp inequalities relating the cardinality of the set \(O(R)\), the Krull dimension, and the number of maximal ideals. A first main result states that for a Prüfer domain \(R\) with finite Krull dimension, \(\dim(R)\), and finite number \(m\) of maximal ideals, \N\[\N(2^{m} + \dim(R)-1)\leq |O(R)|\leq (1 + \dim(R))^{m}\N\]\NA second main result states that for an integrally closed domain \(R\) with finite Krull dimension \(d\geq 1\), and a positive integer \(n\), if \(|O(R)| = n + d\), then the following statements hold true: \((1)\) The number of maximal ideals is finite and satisfies the inequalities: \(\log_{d+1}(n + d) \leq |\mathrm{Max}(R)|\leq \log_{2}(n + 1)\).\N\N\((2)\) \(R\) has exactly \(m = \log_{d+1}(n+d)\) maximal ideals if and only if \(n = (d+1)m-d\) and \(R\) is a Prüfer domain with spectrum isomorphic to a tree consisting of \(m\) chains each containing \(d + 1\) elements and meeting only at their respective minimal elements.\N\N\((3)\) \(R\) has exactly \(m = \log_{2}(n +1)\) maximal ideals if and only if \(n = 2m-1\) and \(R\) is a Prüfer domain with spectrum isomorphic to a tree consisting of a chain containing \(d\) elements and connected at the top element with \(m\) additional edges to each of the \(m\) maximal ideals. Other inequalities related to length, dimension and number of maximal ideals are given.
reviewed by: A. Mimouni
Mathematics Subject Classification ID: 13B02
Mathematics Subject Classification ID: 13B22
Mathematics Subject Classification ID: 13B30
Mathematics Subject Classification ID: 13E99
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13F30
Mathematics Subject Classification ID: 13G05
Mathematics Subject Classification ID: 13H99
zbMATH DE Number: 7857701
zbMATH Keywords: overrings
zbMATH Keywords: Krull dimension
zbMATH Keywords: Prüfer domain
zbMATH Keywords: maximal ideal
MaRDI profile type: Publication
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DOI: 10.1007/S13366-022-00677-5
description: scientific article; zbMATH DE number 7857701