Given any finitely generated subring $R$ of a number field (finite extension of $\mathbb{Q}$) or a global function field (finite extension of $\mathbb{F}_p(T)$), does $R$ have the property that $R/I$ is a finite ring for every non-zero ideal $I$ of $R$?
It is sufficient to show that $R$ has Krull dimension $1$. Indeed, then $R/I$ is a finitely generated artinian ring, which is finite.