# 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 $S$:

$\map V S = \set {\mathfrak p \in \Spec A: \mathfrak p \supseteq S}$

## Also known as

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