Products of commutative rings and zero-dimensionality

Gilmer, Robert; Heinzer, William

doi:10.1090/S0002-9947-1992-1041047-4

Products of commutative rings and zero-dimensionality
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by Robert Gilmer and William Heinzer

Trans. Amer. Math. Soc. 331 (1992), 663-680

DOI: https://doi.org/10.1090/S0002-9947-1992-1041047-4

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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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