uncountable ring with countable quotient ring? Insipired by this post: Countable number of subgroups $\implies $ countable group
I'm wondering if it is possible to have an uncountable commutative ring with identity $R$, such that $R/I$ is countable for some two-sided proper ideal $I$?
It seems to be a really basic problem, but I couldn't give a proof or a counterexample. Any help is really appreciated, thanks!

I believe $R=\Bbb Z\times\Bbb C$ and $I=\{0\}\times\Bbb C$ works.
