Definition:Vanishing Ideal of Set of Prime Ideals

Definition
Let $A$ be a commutative ring with unity.

Let $V \subseteq \operatorname{Spec} A$ be a set of prime ideals of $A$.

Its vanishing ideal is its intersection, the set of elements of $A$ that are in each $\mathfrak p \in V$:
 * $I(V) = \bigcap V$

Also see

 * Definition:Zero Locus of Subset of Ring