Prime spectra in modular varieties
type: Item
instance of: scholarly article
title: Prime spectra in modular varieties
zbMATH Open document ID: 0790.08006
DOI: 10.1007/BF01195383
author: Paolo Agliano
published in: Algebra Universalis
publication date: 1994-06-30T00:00:00Z
review text: Let \(V\) be a congruence modular variety. A congruence \(c\) in \(V\)-algebra \(A\) is prime if \(c\supseteq [a,b]\), \(a,c\in\text{Con}(A)\) implies either \(c\supseteq a\), or \(c\supseteq b\). Denote by \(\text{Spec}(A)\) the set of all prime congruences in \(A\) provided with Zariski topology. It is shown that each irreducible closed subset in \(\text{Spec}(A)\) has a unique generic point. A congruence \(a\) in \(A\) is radical if \(a\) is the intersection of all prime congruences \(b\) in \(A\) such that \(b\supseteq a\). If \(\text{Con}(A)\) satisfies the ascending chain condition on radical congruences, then there exists a commutative ring \(R\) with unit such that \(\text{Spec}(A)\simeq\text{Spec}(R)\) as topological spaces. There is given a characterization of topological spaces \(X\) which are isomorphic to \(\text{Spec}(L)\) for some distributive lattice \(L\). There is also found a Mal'tsev characterization of varieties of algebras in which no nontrivial algebras has a one-element subalgebra. In these varieties the author finds a topological characterization of the spaces \(\text{Spec}(A)\).
Mathematics Subject Classification ID: 08B10
zbMATH DE Number: 488268
zbMATH Keywords: prime spectra
zbMATH Keywords: congruence modular variety
zbMATH Keywords: prime congruences
zbMATH Keywords: Zariski topology
zbMATH Keywords: radical congruences
zbMATH Keywords: topological spaces
reviewed by: Vyacheslav A. Artamonov
MaRDI profile type: Publication
cites work: Q3496382
cites work: A Sheaf-Theoretic Duality Theory for Cylindric Algebras
cites work: Q3775604
cites work: Ideals in universal algebras
cites work: Prime Ideal Structure in Commutative Rings
cites work: Congruences and one element subalgebras
cites work: Q3912631
cites work: Q5678913
full work available at URL: https://doi.org/10.1007/bf01195383
OpenAlex ID: W1995584842
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description: scientific article; zbMATH DE number 488268