Remark

It follows from the preceding proof that the annihilator of $I_n$ in $\widehat{A_p}$ (resp. in $A_p$) is $\mathfrak{p}^n \widehat{A_p}$ (resp. $\mathfrak{p}^n A_p$). Hence the annihilator of the $A$-module $I_n$ is the inverse image in $A$ of the ideal $\mathfrak{p}^n A_p$, which is sometimes denoted by $\mathfrak{p}^{(n)}$ and is called the $n$th symbolic power of the prime ideal $\mathfrak{p}$.