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Intersection of powers of prime ideals

Let $R$ be a Noetherian ring. Let $(x)$ be a prime ideal such that $\bigcap_n (x)^n=0$. Then $R$ is a domain. Is this a known result? I heard its known as the Davis lemma. Can anyone give a ...
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On Prüfer domains

Is there any Prüfer domain $R$ that has a prime ideal $P$ that is not finitely generated but $xP$ is subset of a finitely generated ideal $I$,for some $x$ in $R-P$ and $I$⊂$P$?
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On Prufer domains

Are there any Prufer domains that have an infinity of prime ideals but only one of those primes is not finitely generated?
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Is the ring of all cyclotomic integers a Bezout domain?

My previous question about the theorem (apparently due to Dedekind -- thanks, Arturo Magidin!) that the ring $\overline{\mathbb{Z}}$ of all algebraic integers is a Bezout domain got me thinking about ...