Symbolic power

Summary

Let $P$ be a prime ideal and $n \ge 1$. The $n$-th symbolic power $P^{(n)}$ is the set of ring elements that land in $P^n$ after multiplication by some element outside $P$. It is the smallest primary ideal with radical $P$ that contains $P^n$, so it corrects the fact that $P^n$ itself need not be primary. Symbolic powers appear in the proof of Krull's principal ideal theorem. Over an algebraically closed field, they give the polynomials that vanish to order at least $n$ along a variety.