# Question about proof of Proposition 5.23 in Atiyah-Macdonald

I'm struggling to understand a part of the proof of the following proposition from Atiyah-Macdonald's 'Introduction to Commutative Algebra':

Let $$A \subseteq B$$ be integral domains, $$B$$ finitely generated over $$A$$ (as an $$A$$-module). Let $$v$$ be a non-zero element of B. Then there exists $$u \neq 0$$ in $$A$$ with the following property: any homomorphism $$f$$ of $$A$$ into an algebraically closed field $$\Omega$$ such that $$f(u) \neq 0$$ can be extended to a homomorphism $$g$$ of $$B$$ into $$\Omega$$ such that $$g(v) \neq 0$$.

They treat the case of $$B$$ being generated by a single element $$x$$ over $$A$$ and $$x$$ being algebraic over $$A$$ (i.e. algebraic over the field of fractions of $$A$$). Let $$v = a_0x^n + \dots + a_n$$. Then $$v^{-1}$$ is algebraic over $$A$$, since $$v$$ is a polynomial in $$x$$. Hence there are equations

$$a_0x^m + \dots + a_m = 0\\ a_0'v^{-n} + \dots + a_n' = 0$$

where $$a_i,a_j' \in A$$. Let $$f: A \to \Omega$$ be such that $$f(u) \neq 0$$. Now here is the part that I don't quite understand: They claim that $$f$$ can be extended to a homomorphism $$f_1: A[u^{-1}] \to \Omega$$ by setting $$f_1(u^{-1}) = f(u)^{-1}$$. The hypothesis only states that $$B$$ is finitely generated, not that it's a free $$A$$-module, hence the definition of $$f_1$$ doesn't make sense if there are multiple ways to write elements of $$A[u^{-1}]$$. What am I missing?

$$A[u^{-1}]$$ is naturally isomorphic to the localization of $$A$$ by the multiplicative set $$S=\{1,u,u^2,u^3,...\}$$. Indeed, any element of $$A[u^{-1}]$$ can be written in the form $$\frac{a}{u^n}=au^{-n}$$ for some $$a\in A$$ and $$n\geq 0$$, and two such representations are equal exactly when they are equal in $$S^{-1}A$$.
So now the existence of the extension just follows from the universal property of localization, as the assumption is that $$u$$ (and so all the elements of $$S$$) is mapped to an invertible element of $$\Omega$$.
• I completely forgot about the isomorphism between the localization and $A[u^{-1}]$. Thank you! Apr 11, 2022 at 6:21