Ideal divisibility monoids
type: Item
instance of: scholarly article
title: Ideal divisibility monoids
zbMATH Open document ID: 1015.06019
DOI: 10.1007/BF03322755
author: Bruno Bosbach
published in: Results in Mathematics
publication date: 2003-07-31T00:00:00Z
review text: This paper concerns the role of divisors in abstract ideal theory. The setup is an algebraic multiplicative lattice \(\mathcal A\) (AML) by which here is meant a compactly generated (by a fixed set \({\mathcal A}_c\) of compact elements including 0 and 1) complete lattice with a not necessarily commutative multiplication which is distributive over arbitrary joins and usually has greatest element 1 as a compact multiplicative identity. For elements \(A,B\in L\), \(A\) is a left divisor of \(B\) if \(B = AX\) for some \(X\in L\). A right divisor of \(A\) is defined similarly and \(A\) is a divisor of \(B\) if it is both a left and right divisor of \(A\). And \(A\) is called a left divisor if it left divides each \(B\leq A\). Finally \(\mathcal A\) is called a left divisor AML if each element of \({\mathcal A}_c\) is a left divisor and a left Prüfer AML if each compact element is a left divisor. Thus the lattice of ideals of a Prüfer domain is a Prüfer AML This paper mainly investigates Prüfer AML's and their relationship to the lattice of certain star-ideals (such as \(v\)-ideals and \(t\)-ideals) on monoids.
Mathematics Subject Classification ID: 06F10
Mathematics Subject Classification ID: 13A15
zbMATH DE Number: 1760642
zbMATH Keywords: divisors
zbMATH Keywords: abstract ideal theory
zbMATH Keywords: algebraic multiplicative lattice
zbMATH Keywords: Prüfer domain
zbMATH Keywords: star-ideals
zbMATH Keywords: monoids
reviewed by: D. D. Anderson
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1760642