---
tag: 7VFPQVCNJ7
kind: definition
name: Symbolic power
msc: ["13A15"]
defines: FMPJJBVM5A
uses: [0GYGHKX834, 06HC6HK0Z8, X6JD4AG973, V026AAV8E6]
sources:
  - url: https://en.wikipedia.org/wiki/Symbolic_power_of_an_ideal
---

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]], let $P$ be a [[prime ideal|06HC6HK0Z8]] of $R$, and let $n \ge 1$ be an integer.
The $n$-th *symbolic power* of $P$ is the ideal
$$P^{(n)} = \set{r \in R : sr \in P^n \text{ for some } s \in R \setminus P}.$$

## Remarks

Equivalently, $P^{(n)}$ is the preimage in $R$ of the ideal $P^n R_P$ of the [[localization|X6JD4AG973]] $R_P$.
One always has $P^n \subseteq P^{(n)} \subseteq P$, and $P^{(1)} = P$.
When $R$ is Noetherian, $P^{(n)}$ is the $P$-primary component of $P^n$, and it is the smallest $P$-[[primary ideal|V026AAV8E6]] that contains $P^n$.
Example: in $R = k[x, y, z]/(xy - z^2)$ over a field $k$, with $P = (x, z)$, one has $y \notin P$ and $yx = z^2 \in P^2$, so $x \in P^{(2)}$; but $x \notin P^2$, as $P^2$ is generated by elements of degree $2$.
Non-example: in the same ring, $y \notin P^{(2)}$, since $y \notin P$ and $P^{(2)} \subseteq P$.
For a maximal ideal $\mathfrak{m}$, the power $\mathfrak{m}^n$ is already primary, so $\mathfrak{m}^{(n)} = \mathfrak{m}^n$.

## Explanations

### L1 human

The symbolic power keeps the elements that lie in $P^n$ once one is allowed to clear away factors outside $P$.