Characterization of Prüfer multiplication monoids and domains by means of spectral module systems.
type: Item
instance of: scholarly article
title: Characterization of Prüfer multiplication monoids and domains by means of spectral module systems.
zbMATH Open document ID: 1058.20049
author: Franz Halter-Koch
published in: Monatshefte für Mathematik
publication date: 2003-07-08T00:00:00Z
review text: An integral domain \(A\) is called a Prüfer domain, resp. a Prüfer \(v\)-multiplication domain (PVMD), if every nonzero finitely generated ideal of \(A\) is invertible, resp. every finite type \(t\)-ideal is \(t\)-invertible. It is well-known that there exist several characterizations of the Prüfer domains by properties of their overrings (overring = ring between \(A\) and its quotient field). Most of them have \(t\)-analogs for PVMDs. The aim of the paper under review is to generalize/unify such results in the context of commutative cancellative monoids with zero. Let \(D\) be such a monoid and \(r\) a finitary ideal system on \(D\) (e.g. \(r\) = the \(t\)-system). \(D\) is called an \(r\)-Prüfer monoid if every non-zero \(r\)-finitely generated \(r\)-ideal is \(r\)-invertible. An overmonoid \(T\) of \(D\) is a monoid between \(D\) and its quotient groupoid. \(T\) is said to be \(r\)-linked if \((M\cap D)_r\neq D\) for each \(t\)-maximal \(t\)-ideal \(M\) of \(T\). And \(T\) is said to be pseudo-flat over \(D\) if \(T_M=D_{M\cap D}\) for each \(t\)-maximal \(t\)-ideal \(M\) of \(T\). One of the main results of the paper asserts that a monoid \(D\) is \(r\)-Prüfer iff \(D\) is root-closed and every \(r\)-linked overmonoid of \(D\) is an intersection of quotient monoids of \(D\) iff \(D\) is root-closed and every \(r\)-linked overmonoid of \(D\) is seminormal iff every \(r\)-linked overmonoid of \(D\) is pseudo-flat over \(D\).
reviewed by: Tiberiu Dumitrescu
Mathematics Subject Classification ID: 20M14
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 20M12
Mathematics Subject Classification ID: 13G05
zbMATH DE Number: 1947311
zbMATH Keywords: Prüfer domains
zbMATH Keywords: Prüfer monoids
zbMATH Keywords: finitely generated ideals
zbMATH Keywords: invertible ideals
zbMATH Keywords: commutative cancellative monoids
zbMATH Keywords: ideal systems
MaRDI profile type: Publication
full work available at URL: s00605-002-0496-x
OpenAlex ID: W1981105323
DOI: 10.1007/S00605-002-0496-X
description: scientific article; zbMATH DE number 1947311