Ultraproducts of PM-rings and MP-rings
type: Item
instance of: scholarly article
title: Ultraproducts of PM-rings and MP-rings
zbMATH Open document ID: 0539.13003
DOI: 10.1016/0022-4049(84)90010-0
published in: Journal of Pure and Applied Algebra
publication date: 1984-01-01T00:00:00Z
review text: The author considers properties of commutative rings preserved under ultraproducts. Since an ultraproduct is a quotient of a direct product it is sufficient to treat these two constructions separately. The properties considered are PM (every prime is contained in a unique maximal), MP (every prime contains a unique minimal), soft (Hausdorff maximal spectrum and zero Jacobson radical), TB (maximal spectrum homeomorphic to Boolean spectrum), weak Baer (element annihilators are idempotent generated principal), and Baer (ideal annihilators are idempotent generated principal). All are shown to be preserved by ultra-products for Baer and MP, and the latter is preserved for reduced rings. One technical tool, used in the Baer result, characterizes the incomplete ultraproducts of Boolean algebras.
Mathematics Subject Classification ID: 13A99
Mathematics Subject Classification ID: 12L10
Mathematics Subject Classification ID: 13L05
zbMATH DE Number: 3857250
zbMATH Keywords: ultraproduct
zbMATH Keywords: PM
zbMATH Keywords: MP
zbMATH Keywords: weak Baer
author: Maria Contessa
reviewed by: Andy R. Magid
MaRDI profile type: Publication
full work available at URL: 0022-4049(84)90010-0
OpenAlex ID: W2020220794
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description: scientific article; zbMATH DE number 3857250