TITLE: Example of a commutative noetherian ring with $1$ which is neither domain nor local and has a principal prime ideal of height $1.$
QUESTION [6 upvotes]: I am trying to construct an example of a ring satisfying the followings. 

A commutative noetherian ring with $1$ which is neither domain nor local and has a principal prime ideal of height $1.$ 

I know that a local noetherian ring having a height $1$ principal prime ideal is a domain. Actually I wanted to prove this without the local condition. I couldn't prove this hence I am looking for a counterexample. I need some help. Thanks.

REPLY [4 votes]: Let $k$ be a field and $A=k[x,y]/(xy)$.  Then the ideal $(y-1)\subset A$ is a principal height $1$ prime.  (Note that this ring is not local, and the non-localness is essential to the example in that if you localize at $(y-1)$ then the ring becomes a domain.)
As a hint to what's going on here that can't happen in the local case, notice that $(y-1)x=-x$, so $x$ is divisible by $y-1$ arbitrarily many times.  In a local Noetherian ring this would imply $x=0$ by the Krull intersection theorem.

REPLY [4 votes]: Let $R$ be a noetherian ring with principal prime ideal $P$ of height $1$.
Then $R\times R$ is Noetherian, non local, not a domain, and still has a principal prime ideal of height $1$: $P\times R$.