How does the cardinality of the maximal spectrum relate to the cardinality of the ring? I have an integral domain $R$, not a field, of finite character (each nonzero element of the ring is contained in only finitely many maximal ideals). If the number of maximal ideals of $R$ is infinite, must this cardinal be less than or equal to the cardinality of $R?$ I know the cardinals can be equal: if $k$ is an infinite field, the maximal ideals of $k[x,y]$ are in one-to-one correspondence with the set of points $(a,b) \in k \times k,$ and the cardinalities of $k,$ $k \times k,$ and $k[x,y]$ are equal. Can the number of maximal ideals ever be greater than the cardinality of $R?$
 A: If $R$ is infinite and each nonzero element of $R$ is contained in only finitely many maximal ideals of $R$, then there can be at most $\aleph_0\cdot |R|=|R|$ different maximal ideals, since there are less than $\aleph_0$ maximal ideals for each nonzero element of $R$.
In general, if $R$ is infinite the set of maximal ideals of such a ring $R$ could have any positive cardinality less than or equal to $|R|$.  For instance, let $k$ be a field of a given infinite cardinality $\kappa$ and let $S$ be the set of irreducible monic polynomials in $k[x]$.  Then $|S|\geq |k|=\kappa$ since $x-a\in S$ for each $a\in k$ and $|S|\leq |k[x]|=\kappa$ so $|S|=\kappa$.  Now for any cardinal $\lambda$ such that $0<\lambda\leq\kappa$, let $T\subseteq S$ be a subset of cardinality $\lambda$ and let $R=k[x][(S\setminus T)^{-1}]$ be the localization of $k[x]$ which inverts each element of $S\setminus T$.  Then $R$ has cardinality $\kappa$ and has $\lambda$ maximal ideals since its maximal ideals are in bijection with $T$.  Since $R$ is a PID, it is of finite character.
