Lemma 2

Tag 04VV, permanent. Cite it as [Bourbaki, Tag 04VV].

Commutative Algebra, chapter VI, § 7. Approximation theorem

Let A be a valuation ring and b an ideal of A distinct from A. Then the radical c of b is a prime ideal.

Suppose that $xy \in c$ . Then there exists $n \geq 1$ such that $(xy)^n \in b$ . Let $v$ be a valuation associated with A. If, for example, $v(x) \geq v(y)$ , then

v(x^{2n}) \geq v(x^n y^n),

whence $x^{2n} \in b$ and $x \in c$ .

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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