The function field of an irreducible classical affine variety

Assume AC for the domain-coordinate interface used here. For a classical affine variety $X$, its function field is $k(X)=\mathrm{Frac}(k[X])\mathrm{.}$ The coordinate ring is a nonzero integral domain by A classical affine variety has a domain coordinate ring, and conversely, so The field of fractions $\mathrm{Frac}(D)=(D\setminus \{0\}{)}^{-1}D$ of an integral domain and $\mathrm{Frac}(D)$ is a field and $d\mapsto d\mathrm{/}1$ embeds the integral domain $D$ give a field and the injective map $a\mapsto a\mathrm{/}1$. Elements have the form $a\mathrm{/}b$ with $b\mathrm{\ne }0$; they are called rational functions. Equality is $a\mathrm{/}b=c\mathrm{/}d$ iff $ad=bc$. Fractions are field elements, not initially functions defined at every point.

Source comparison: Milne, Algebraic Geometry, v6.10, §2i p. 49 and §3k p. 74. Conventions here distinguish arbitrary affine algebraic sets from nonempty irreducible varieties.