Commutative Rings with a Prescribed Number of Isomorphism Classes of Minimal Ring Extensions

David E. Dobbs ()
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David E. Dobbs: University of Tennessee, Department of Mathematics

A chapter in Rings, Polynomials, and Modules, 2017, pp 145-158 from Springer

Abstract: Abstract Let κ be a cardinal number. If κ ≥ 2, then there exists a (commutative unital) ring A such that the set of A-algebra isomorphism classes of minimal ring extensions of A has cardinality κ. The preceding statement fails for κ = 1 and, if A must be nonzero, it also fails for κ = 0. If κ ≤ ℵ 0 $$ \kappa \leq \aleph _{0} $$ , then there exists a ring whose set of maximal (unital) subrings has cardinality κ. If an infinite cardinal number κ is of the form κ = 2 λ for some (infinite) cardinal number λ, then there exists a field whose set of maximal subrings has cardinality κ.

Keywords: Commutative ring; Minimal ring extension; Cardinal number; Polynomial ring; Prime ideal; Valuation domain; Maximal subring; Idealization; Ordinal number; Primary: 13B99; Secondary: 13A15 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/978-3-319-65874-2_8

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