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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Products of commutative rings and zero-dimensionality


Authors: Robert Gilmer and William Heinzer
Journal: Trans. Amer. Math. Soc. 331 (1992), 663-680
MSC: Primary 13C15; Secondary 13E10
MathSciNet review: 1041047
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Abstract: If $ R$ is a Noetherian ring and $ n$ is a positive integer, then there are only finitely many ideals $ I$ of $ R$ such that the residue class ring $ R/I$ has cardinality $ \leq n$. If $ R$ has Noetherian spectrum, then the preceding statement holds for prime ideals of $ R$. Motivated by this, we consider the dimension of an infinite product of zero-dimensional commutative rings. Such a product must be either zero-dimensional or infinite-dimensional. We consider the structure of rings for which each subring is zero-dimensional and properties of rings that are directed union of Artinian subrings. Necessary and sufficient conditions are given in order that an infinite product of zero-dimensional rings be a directed union of Artinian subrings.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1992-1041047-4
PII: S 0002-9947(1992)1041047-4
Keywords: Products of commutative rings, zero-dimensionality, ideals of finite norm, directed union of Artinian subrings, Noetherian ring, Noetherian spectrum, finite spectrum, hereditarily Noetherian ring, Krull dimension
Article copyright: © Copyright 1992 American Mathematical Society