Definition:Vanishing Set of Subset of Ring

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

Let $S \subseteq A$ be a subset.

The vanishing set of $S$ is the set of prime ideals of $A$ containing $S$:
 * $V(S) = \{\mathfrak p \in \operatorname{Spec}(A) : \mathfrak p \supseteq S \}$

Also known as
The vanishing set of $S$ is also known as the zero locus of $S$.

Also see

 * Definition:Zariski Topology on Prime Spectrum of Ring
 * Definition:Vanishing Ideal of Set of Prime Ideals