Symbolic power

Remarks

Equivalently, $P^{(n)}$ is the preimage in $R$ of the ideal $P^n R_P$ of the localization $R_P$. 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 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 $\mathfrak{m}^{(n)} = \mathfrak{m}^n$.