For an ideal II in a ring RR, recall that D(I)⊆Spec⁡(R)D(I)\subseteq \operatorname{Spec}(R) denotes the set of prime ideals ℘\wp of RR not containing II, i.e. primes ℘\wp such that (R−℘)∩I≠∅(R-\wp)\cap I\neq \emptyset. Alternatively, you can think of D(I)D(I) as the set of primes ℘\wp for which there exists an element f∈If\in I that is nonzero modulo ℘\wp.

Let I,JI,J be two ideals of RR. What are D(I)∩D(J)D(I)\cap D(J) and D(I)∪D(J)D(I)\cup D(J)?

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