Lattices of two-sided ideals of locally matricial algebras and the \(\Gamma\)-invariant problem
type: Item
instance of: scholarly article
title: Lattices of two-sided ideals of locally matricial algebras and the \(\Gamma\)-invariant problem
zbMATH Open document ID: 1057.06004
DOI: 10.1007/BF02771525
published in: Israel Journal of Mathematics
publication date: 2004-10-15T00:00:00Z
review text: It is well known that the lattice of two-sided ideals of a von Neumann regular ring is distributive. G. M. Bergman proved that every algebraic distributive lattice is either isomorphic to the lattice of lower subsets of a partially ordered set or, with at most countably many compact elements, is isomorphic to the two-sided ideal lattice of a locally matricial algebra. In contrast, \textit{F. Wehrung} [Isr. J. Math. 103, 177--206 (1998; Zbl 0916.06018)] constructed an algebraic distributive lattice with \(\aleph_2\) compact elements which cannot be realized as the lattice of two-sided ideals of any von Neumann regular ring. Further, he proved that if an algebraic distributive lattice has \(\aleph_1\) compact elements, then it can be realized as the lattice of two-sided ideals of a von Neumannr regular ring; however he proved recently that the result fails for locally matricial algebras. The main result of the present paper is the representation of every algebraic distributive lattice whose compact elements form a lattice as the lattice of a locally matricial algebra.
reviewed by: Radomír Halaš
Mathematics Subject Classification ID: 06D05
Mathematics Subject Classification ID: 06A12
zbMATH DE Number: 2108111
zbMATH Keywords: representation
zbMATH Keywords: locally matricial algebra
zbMATH Keywords: \(\Gamma\)-invariant problem
zbMATH Keywords: algebraic distributive lattice
author: Pavel Ružička
MaRDI profile type: Publication
Wikidata QID: Q114586130
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description: scientific article; zbMATH DE number 2108111