Tidy Noether lattices
type: Item
instance of: scholarly article
title: Tidy Noether lattices
zbMATH Open document ID: 0632.06021
DOI: 10.1007/BF01188383
author: Linda Becerra
published in: Algebra Universalis
publication date: 1987-01-01T00:00:00Z
review text: A Noether lattice \({\mathcal L}\) satisfying the union condition on primes which is not a domain and in which every nonzero principal element is integrally closed is characterized in terms of its direct summands. It is shown that either: \((1)\quad if\) \({\mathcal L}\) has no proper nonzero direct summands, then every nonzero principal element of \({\mathcal L}\) is integrally closed if and only if \({\mathcal L}\) is a local Noether lattice whose maximal element is principal and has square zero; or \((2)\quad if\) \({\mathcal L}\) has a proper nonzero direct summand, then every nonzero principal element of \({\mathcal L}\) is integrally closed if and only if for each minimal direct summand A of \({\mathcal L}\), the quotient lattice [0,A] is an integrally closed domain.
Mathematics Subject Classification ID: 06F10
Mathematics Subject Classification ID: 13C05
Mathematics Subject Classification ID: 13A15
zbMATH DE Number: 4027542
zbMATH Keywords: union condition on primes
zbMATH Keywords: principal element
zbMATH Keywords: direct summands
zbMATH Keywords: integrally closed
zbMATH Keywords: local Noether lattice
zbMATH Keywords: maximal element
zbMATH Keywords: quotient lattice
MaRDI profile type: Publication
cites work: Structure theorems for regular local Noether lattices
cites work: Q5659674
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cites work: A-Transforms and Hilbert Functions in Local Lattices
cites work: Prime sequences and distributivity in local Noether lattices
cites work: Structure and embedding theorems for unique normal decomposition lattices
cites work: Asymptotic and integral closure of elements in multiplicative lattices
full work available at URL: https://doi.org/10.1007/bf01188383
OpenAlex ID: W2067516383
description: scientific article; zbMATH DE number 4027542