The coordinate ring of a classical affine algebraic set

For any affine algebraic set $X\subseteq k^n$, its coordinate ring is $k[X]=k[x_1,\dots ,x_n]\mathrm{/}I(X)\mathrm{.}$ Write ${\bar{x}}_i$ for the coordinate classes. The quotient operations and identity are those of The quotient ring $R\mathrm{/}I$ with $(r+I)(s+I)=rs+I$ and For a two-sided ideal $I$, the additive cosets form a ring $R\mathrm{/}I$ with identity $1+I$. The structure map sends $c\in k$ to its constant class; all $k$-algebras and maps are unital, with the zero algebra allowed. Thus $k[\mathrm{\varnothing }]=0$, where $0=1$. Since $I(X)$ is radical by The classical vanishing ideal, $k[X]$ is reduced: ${\bar{f}}^m=0$ implies $f^m\in I(X)$ and hence $f\in I(X)$, so $\bar{f}=0$. The finite coordinate classes generate it as a $k$-algebra.

Source comparison: Milne, Algebraic Geometry, v6.10, §2i p. 48. Conventions here distinguish arbitrary affine algebraic sets from nonempty irreducible varieties.