TITLE: Weil divisors on non Noetherian schemes
QUESTION [6 upvotes]: Let X be an integral scheme that is separated (say over an affine scheme). Define a Weil divisor as a finite integral combination of height 1 points of X, where the height of a point of X is the dimension of its local ring.  Let Z be a closed subscheme of X, Z not equal to X.  
Is there an example of such an X and Z where Z contains infinitely many Weil divisors?

REPLY [3 votes]: Let R be the integral closure of $\mathbb{Z}$ in the algebraic closure of $\mathbb{Q}$.
Then $\dim (R)=1$ and there are infinitely many prime ideals lying above a prime ideal
$p\mathbb{Z}$. Hence the closed set $V(pR)$ has the required property.