Regular elements in multiplicative lattices
type: Item
instance of: scholarly article
title: Regular elements in multiplicative lattices
zbMATH Open document ID: 1168.06008
author: C. Jayaram
published in: Algebra Universalis
publication date: 2009-01-29T00:00:00Z
review text: Throughout, \(L\) is a not necessarily modular multiplicative lattice (\(L\) has a commutative associative product that distributes over joins and has greatest element 1 as a multiplicative identity) that is join-generated by a multiplicatively closed set of compact elements with 1 compact and join-generated by a set of principal elements (\(e\) is called principal if \(a\wedge be = ((a : e)\wedge b)e\) and \((ae\vee b): e=(b:e)\vee a)\). An element \(a\) is called invertible if \(a\) is principal and \(0 : a = 0\), and \(a\) is called regular if it contains an invertible element. The paper gives a number of characterizations of the lattices \(L\) in which every regular element is invertible, such as every regular element is a product of prime elements, or every invertible element is contained in only finitely many prime elements, and for every regular maximal prime \(m\) of \(L\), \(m\) is locally principal, and \(\bigwedge_{k=1}^\infty m^k\) is nonregular. Several new characterization of Dedekind lattices (every element not contained in a minimal prime element is weak meet principal (i.e., satisfies the first of the two principal element identities)) and WI-lattices (every compact element \(a\in L\) is principal and \((0 : (0 : a))\vee (0 : a) = 1)\) are given. Applications are given to commutative rings.
reviewed by: D. D. Anderson
Mathematics Subject Classification ID: 06F10
Mathematics Subject Classification ID: 06F05
Mathematics Subject Classification ID: 13A15
zbMATH DE Number: 5500517
zbMATH Keywords: C-lattice
zbMATH Keywords: principal element
zbMATH Keywords: invertible element
MaRDI profile type: Publication
full work available at URL: s00012-008-2066-0
OpenAlex ID: W2034617739
DOI: 10.1007/S00012-008-2066-0
description: scientific article; zbMATH DE number 5500517