# Relating the cardinalities of rings and their spectra

Does there exist an uncountable (possibly commutative, unital) ring with a countably infinite spectrum?

More generally, given two cardinalities $\kappa$ and $\lambda$, does there exist a ring $R$ such that $|R| = \kappa$ and $|Spec(R)| = \lambda$?

Other than the obvious restrictions (such as $\lambda < 2^{\kappa}$ and some number-theoretic restrictions if $\kappa$ and $\lambda$ are both finite), I haven't been able to come up with any limitations on $\kappa$ and $\lambda$, but haven't found any examples of answers to my first question, either.

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Don't you mean $\lambda\leq 2^\kappa$, rather than $\lambda\lt 2^\kappa$? – Joel David Hamkins May 11 '13 at 1:03
For $\kappa$ and $\lambda$ infinite (and cardinal arithmetic sufficiently well-behaved), take the product of a field of cardinality $\kappa$ and a polynomial ring in one variable over a field with cardinality $\lambda$. – Will Sawin May 11 '13 at 5:48