Doubt in Atiyah-Macdonald proposition 5.13

This is proposition 5.13 from Atiyah-Macdonald:
Let $$A$$ be an integral domain. Then the following are equivalent:

• $$A$$ is integrally closed
• $$A_{\mathfrak p}$$ is integrally closed, for each prime ideal $$\mathfrak p$$
• $$A_{\mathfrak m}$$ is integrally closed, for each maximal ideal $$\mathfrak m$$

This is the proof given in the text:
Let $$K$$ be the field of fractions of $$A$$, let $$C$$ be the integral closure of $$A$$ in $$K$$, and let $$f: A \to C$$ be the identity mapping of $$A$$ into $$C$$. Then $$A$$ is integrally closed iff $$f$$ is surjective, and by (5.12) $$A_{\mathfrak p}$$ (resp. $$A_{\mathfrak m}$$) is integrally closed iff $$f_{\mathfrak p}$$ (resp.$$f_{\mathfrak m}$$) is surjective. Now use (3.9). $$\blacksquare$$

Thm 5.12) Let $$A \subset B$$ be rings, $$C$$ the integral closure of $$A$$ in $$B$$. Let $$S$$ be a multiplicatively closed subset of $$A$$. Then $$S^{-1}C$$ is the integral closure of $$S^{-1}A$$ in $$S^{-1}B$$.

Thm 3.9) Let $$\phi: M \to N$$ be an $$A$$-module homomorphism. Then the following are equivalent:

• $$\phi$$ is surjective
• $$\phi_{\mathfrak p}:M_{\mathfrak p} \to N_{\mathfrak p}$$ is surjective for each prime ideal $$\mathfrak p$$
• $$\phi_{\mathfrak m}:M_{\mathfrak m} \to N_{\mathfrak m}$$ is surjective for each maximal ideal $$\mathfrak m$$

I am not able to see where we have used the hypothesis that $$A$$ is an integral domain. I'm pretty sure that I'm missing something. Please can somebody guide me.

• @K02: Atiyah and MacDonald use a generalised definition of the ring of fractions that applies to any commutative ring. However, the definition they give for integrally closed is restricted to integral domains (and uses the term "field of fractions" because for integral domains, you know the ring of fractions is a field.) Nov 27, 2023 at 20:52