I have to prove that $R=\mathbb{Z}[x,y]/(xy-7)$ is regular. So I have to prove that $R$ is local, Noetherian and that $$dim R= dim_{R \setminus P} P/P²$$ where $P$ is the maximal ideal of $R$.
$R$ is Noetherian because $\mathbb{Z}$ is Noetherian, but why is it a local ring? And how can I prove that $$dim R \geq dim dim_{R \setminus P} P/P² ?$$