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.