Prime ideals $\mathfrak{p} \supset \mathfrak{a}$ are finite in one-dimensional Noetherian domain

Question

Let $A$ be a one-dimensional Noetherian domain. Let $\mathfrak{a} \neq 0$ be an ideal of $A$. How do I prove that prime ideals $\mathfrak{p} \supset \mathfrak{a}$ are finite?

Thanks.

Answer 1

The quotient ring $A/\mathfrak a$ is zero-dimensional and noetherian, so it is artinian. But artinian rings have finitely many maximal ideals.