Let $(R, mathfrak m)$ be a Noetherian local ring. Let P be a property of $R$. Set

$$ P(R) ={mathfrak p in Spec(R),,, |,,, R_{mathfrak p}, , mbox{is } P},$$

$$ nP(R) ={mathfrak p in Spec(R),,, |,,, R_{mathfrak p}, , mbox{is not} P}.$$

I knew that:

(1) If $R$ is excellent ring then $P(R)$ is open, where P is "regular";

(2) If $R$ is homomorphic image of a Gorenstein ring then $P(R)$ is open, where P is "Gorenstein".

The methods: Using Topological Nagata criterion and Ring theoretic Nagata criterion.

In these cases $nP(R)$ is closed.

**My question:** Find $I$, $J$ so that $nP(R)=Var(I)$ in Case 1 and $nP(R)=Var(J)$ in Case 2.

Thank you for your comment or answer for me!