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

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 \( \kappa \leq \aleph _{0} \), 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.