The Stacks project

Lemma 15.125.6. Let $(A, \mathfrak m, \kappa )$ be a Noetherian normal local domain of dimension $2$. If $a \in \mathfrak m$ is nonzero, then there exists an element $c \in A$ such that $A/cA$ is reduced and such that $a$ divides $c^ n$ for some $n$.

Proof. Let $\text{div}(a) = \sum _{i = 1, \ldots , r} n_ i \mathfrak p_ i$ with notation as in the proof of Lemma 15.125.5. Choose $c \in \mathfrak p_1 \cap \ldots \cap \mathfrak p_ r$ with $A/cA$ reduced, see Lemma 15.125.5. For $n \geq \max (n_ i)$ we see that $-\text{div}(a) + \text{div}(c^ n)$ is an effective divisor (all coefficients nonnegative). Thus $c^ n/a \in A$ by Algebra, Lemma 10.157.6. $\square$

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