Let $K$ be a number field with ring of integers $O_K$. Does there exist a $2$-dimensional subring $A\subset O_K$?
Clearly, if such a subring $A\subset O_K$ exists, we have that $A$ is an integral domain. Also, taking the integral closure (i.e. the normalization) we obtain a normal integral domain $A\subset O_K$ of dimension two.
Two problems remain:
How do you show that there exists such a subring?
How do you get a noetherian subring?
Also, I was wondering whether there is a "canonical" choice for such an $A$ (if it exists).