Is $Spec(\Bbb R[x,y]/\langle xy\rangle)[ \frac{1}{ x−y} ]$ connected?

Question

I cannot see if $Spec(\Bbb R[x,y]/\langle xy\rangle)[ \frac{1}{ x−y} ]$ is connected.

I have tried to see if there exists some nontrivial idempotent in the ring $\Bbb R[x,y]/\langle xy\rangle)[ \frac{1}{ x−y} ]$ it seems not easy. I have just proved that we have no nontrivial idempotent in $\Bbb R[x,y]/\langle xy\rangle$ and I guess it may helps.

Any ideas of how to see if this affine scheme is connected?

Thanks so much!

Answer 1

It is disconnected. Let $I=⟨x⟩,J=⟨y⟩$ then $1=zx−zy∈I+J$ so $I+J=\langle 1\rangle$. And $IJ=⟨xy⟩=⟨0⟩$ so $D_I∪D_J=D_{I+J}=D_{⟨1⟩}=X$,$D_I∩D_J=D_{IJ}=D_{⟨0⟩}=∅$.

This proves that the scheme is disconnected.