# Definition:Vanishing Set of Subset of Ring

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## 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 $S$.