Symbolic power of an ideal

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

$P^{(n)}=P^{n}R_{P}\cap R=\{f\in R\mid sf\in P^{n}{\text{ for some }}s\in R-P\}.$

It is the smallest P-primary ideal containing the n-th power Pⁿ. 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 Pⁿ. We have: $P^{(1)}=P$ and if P is a maximal ideal, then $P^{(n)}=P^{n}$.

References

• http://www.math.lsa.umich.edu/~hochster/711F07/L09.07.pdf