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Here is the excerpt of the book where I suspect a mistake (page 66):

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Where they say "The restriction to $A$ of the natural homomorphism $A^\prime \to k^\prime$" I think we don't want a restriction. We start with the quotient map $\pi: A[x^{-1}] \to A[x^{-1}] /m$ where $m$ is a maximal ideal containing $x^{-1}$. We take an algebraic closure $\Omega$ of the field $A[x^{-1}] /m$ and consider the map $i \circ \pi: A[x^{-1}] \to \Omega$. Then by the previous theorem, (5.21), we can extend $i \circ \pi$ to some valuation ring $B$ of $K$ containing $A[x^{-1}]$: $g: B \to \Omega$ such that $g|_{A[x^{-1}]} = i \circ \pi$. Then $g(x^{-1}) = 0$. Hence $x^{-1} \in ker(g)$ and since the kernel is a proper ideal of $B$, $x^{-1}$ is not a unit in $B$ and hence $x$ is not in $B$.

Do you agree with my version and that what is written in Atiyah-Macdonald is not correct? Thank you.

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    @BruceEvans Thank you!2012-07-18

1 Answers 1

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Congratulations for spotting the difficulty and correcting it, Clark: you are absolutely right and I completely agree with your version!

As a slightly different formulation for the proof that $x\notin B$, I would just remark that if we had $x\in B$, we would deduce the absurd conclusion $1=g(1)=g(x\cdot x^{-1})=g(x)\cdot g(x^{-1})=g(x)\cdot 0=0$

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    Thanks to you and @MattN, then.2012-07-18