Does every Noetherian domain have finitely many height 1 prime ideals? 
Let $A$ be a Noetherian domain. Is the set $\{P\subset A \mid P \mbox{ prime ideal, } \dim A_P=1\}$ always finite?

I can prove for $f \neq 0, f\in A$, the set $\{P\subset A \mid \dim A_P=1, f\in P\}$ is finite (by using the primary decomposition of $\sqrt{(f)}$). The above statement is just the case when $f=0$.

No. Look at the ring $A=\mathbb{Z}$.
