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

P^{(n)} = P^n R_P \cap R = \{ f \in R \mid sf \in P^n \text{ for some }s \in R - P \}.[1]

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.


  1. Here, by abuse of notation, we write I \cap R to mean the pre-image of I along the localization map R \to R_P.

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