Subdirect decomposition of rings and the axiom of choice
type: Item
instance of: scholarly article
title: Subdirect decomposition of rings and the axiom of choice
zbMATH Open document ID: 0628.03038
DOI: 10.1007/BF01206469
author: Yehuda Rav
published in: Archiv der Mathematik
publication date: 1988-01-01T00:00:00Z
review text: It is shown that the following is a theorem of ZF set theory without choice: A commutative ring R admits a representation as a subdirect product of subdirectly irreducible rings if and only if R contains a maximal ideal. It follows then from a theorem of \textit{W. Hodges} [J. Lond. Math. Soc., II. Ser. 19, 285-287 (1979; Zbl 0403.03037)] that already for the class of unique factorization domains, the subdirect product representability by subdirect irreducibles implies the axiom of choice. This answers a question of \textit{G. Grätzer} [Universal algebra, 2nd edition (1979; Zbl 0412.08001), p. 161, Problem 30]. We also draw attention to the little known fact that the very notion and terminology of subdirect product representation by subdirect irreducibles were invented by \textit{R. Remak} in 1930 [cf. J. Reine Angew. Math. 163, 1-44 (1930); Jahrb. Fortschr. Math. 56, I, p. 129].
Mathematics Subject Classification ID: 03E25
Mathematics Subject Classification ID: 13A99
Mathematics Subject Classification ID: 08A30
zbMATH DE Number: 4021130
zbMATH Keywords: ZF set theory
zbMATH Keywords: commutative ring
zbMATH Keywords: subdirect product
zbMATH Keywords: subdirectly irreducible rings
zbMATH Keywords: maximal ideal
zbMATH Keywords: axiom of choice
Wikidata QID: Q114694019
MaRDI profile type: Publication
cites work: Subdirect unions in universal algebra
cites work: Q5544351
cites work: Global subdirect products
cites work: Subdirectly irreducible commutative rings
cites work: Q3872514
cites work: Q4479671
full work available at URL: https://doi.org/10.1007/bf01206469
OpenAlex ID: W2010190587
description: scientific article; zbMATH DE number 4021130