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Products of Commutative Rings and Zero-Dimensionality
Robert Gilmer and William Heinzer
Transactions of the American Mathematical Society
Vol. 331, No. 2 (Jun., 1992), pp. 663-680
Published by: American Mathematical Society
Stable URL: http://www.jstor.org/stable/2154134
Page Count: 18
You can always find the topics here!Topics: Subrings, Mathematical rings, Algebra, Polynomials, Integers, Mathematical theorems, Direct products, Cardinality, Isomorphism
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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 ≤ 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.
Transactions of the American Mathematical Society © 1992 American Mathematical Society