TITLE: An integral domain that is Noetherian, integrally closed, but not one-dimensional
QUESTION [7 upvotes]: A Dededind domain is defined as an integral domain which is integrally closed, one-dimensional, and Noetherian.
Also I know an equivalent characterization that a domain is Dedekidn if and only if every nonzero ideals can factor into prime ideals.
Thus, can anyone provide me an integral domain that is Noetherian, integrally closed, but not one-dimensional, and show an ideal that cannot factor into primes and/or does not own unique factorization?

REPLY [2 votes]: Here is a geometric example. We consider the affine quadric hypersurface  $ \Bbb{C}^3$ corresponding to $\Bbb{C}[x,y,z]/(xy - z^2)$. By Hartshorne II Ex. 6.4 the ring is integrally closed. If you really want to show unique factorization, follow the method of proof of Alex's comment above and show $(z^2,x,y)$ does not have unique factorization.