In a number ring, every non-zero ideal has finite index.
In order to build a better intuition on what makes number rings special to have this property, I'm trying to find some "minimal" counterexamples to this property. A number ring is a subring of a finite extension of $\mathbb{Q}$. So, specifically, I'm trying to find examples of a non-zero ideal $I$ of a subring $R$ of a finite field extension $E/F$ such that $R/I$ is infinite.
I think such an example would be $F = k(X)$ the field of rational functions over an infinite field $k$, $E$ any finite extension of $F$ (for instance, $E = F$), $R = k[X]$ and $I = (X) \triangleleft R$.
By the contrapositive of this statement, the following are classes of examples:
- Non-Noetherian domains
- Domains with dimension greater than 1
What would be some other examples?
