# Symbolic power of a prime ideal

In algebra, given a ring *R* and a prime ideal *P* in it, the ** n-th symbolic power** of

*P*is the ideal

^{[1]}

It is the smallest *P*-primary ideal containing the *n*-th power *P*^{n}. Very roughly, it consists of functions with zeros of order *n* along the variety defined by *P*. If *R* is Noetherian, then it is the *P*-primary component in the primary decomposition of *P*^{n}. We have: and if *P* is a maximal ideal, then .

## References

- ↑ Here, by abuse of notation, we write to mean the pre-image of
*I*along the localization map .

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