In [THIS PROBLEM](https://mathoverflow.net/questions/118258/maximal-ideals-in-rx/118311) it is proved that for any non-zero ring $R$ with identity, $R[x]$ has an infinite number of maximal left ideals. Is it possible for an uncountable non-zero ring $R$ with identity, $R[x]$ has only a countable number of maximal left ideals ?? 



Number of Maximal Left Ideals 
Generalizing the answer by wccanard: Let $R$ be a commutative ring. Then the kernel of $R[x] \twoheadrightarrow R\_{red}[x]$ consists of nilpotent elements, hence this map induces a *homeomorphism* $\mathrm{Spec}(R\_{red}[x]) \cong \mathrm{Spec}(R[x])$. It restricts to a homeomorphism between the subspaces of closed points $\mathrm{Spm}(R\_{red}[x]) \cong \mathrm{Spm}(R[x])$. Now take any uncountable $R$ such that $R\_{red}$ is a finite field.


