Some question of codimension 1 (1) "For affine variety  $V$ of $\mathbb{A}^{n}$ such that its coordinate ring is UFD,
     closed subvariety of $V$ which has codimension 1 is cut out by a single equation."
I looked at the proof of this statement, Where I have been using UFD in I do not know...
(2) I want to see proof of follwing statement....
"Any closed subvariety of affine normal variety with codimension 1 is cut out by a single equation."

As for (1): a closed subvariety of $V$ is defined by a prime ideal $p$ of the coordinate ring $k[V]$ of height $1$. If $k[V]$ is a UFD such a prime ideal is principal.
As for (2): the statement is false. The closed subvariety is cut out by a single equation locally but not globally as in the case of a UFD.
