Ideal semigroups of noetherian domains and Ponizovski decompositions
type: Item
instance of: scholarly article
title: Ideal semigroups of noetherian domains and Ponizovski decompositions
zbMATH Open document ID: 1114.13015
author: Franz Halter-Koch
published in: Journal of Pure and Applied Algebra
publication date: 2007-03-27T00:00:00Z
review text: Let \(R\) be a noetherian integral domain, and \(L\) a field, which is a finite extension of the field \(K\) of quotients of \(R\). The author studies the multiplicative semigroup \({\mathcal F}_L(R)\) of all \(R\)-lattices contained in \(L\) (i.e., finitely generated \(R\)-modules lying in \(L\), and containing a \(K\)-basis of \(L\)). After showing that idempotents of \({\mathcal F}_L(R)\) coincide with \(R\)-orders contained in \(L\) it is proved that if \(R\) is one-dimensional, then \({\mathcal F}_L(R)\) is complete, and if \(R\) is Dedekind, then \({\mathcal F}_L(R)\) is a Clifford semigroup if and only if the extension \(L/K\) is quadratic. Two \(R\)-lattices \(A,B\in {\mathcal F}_L(R)\) are equivalent, if for some non-zero \(\lambda\in L\) one has \(A=\lambda B\), and the set of resulting equivalence classes forms a semigroup \(S_L(R)\). The author describes its structure in terms of its Ponizovski factors, which the author defines for arbitrary semigroup \(S\), by putting for every idempotent \(e\in S\) \[ P_e=Se/\bigcup_fSf, \] where \(f\) runs over all idempotents of \(S\) satisfying \(ef=e\) and \(f\neq e\). The Ponizovski factors are also used to give simple characterizations of Clifford semigroups and complete semigroups.
reviewed by: Władysław Narkiewicz
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 11R04
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 20M14
zbMATH DE Number: 5137677
zbMATH Keywords: \(R\)-lattices
zbMATH Keywords: noetherian domains
zbMATH Keywords: Clifford semigroups
zbMATH Keywords: complete semigroups
MaRDI profile type: Publication
full work available at URL: j.jpaa.2006.07.012
OpenAlex ID: W1969341368
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DOI: 10.1016/J.JPAA.2006.07.012
description: scientific article; zbMATH DE number 5137677