Chapter 1, Exercise 1.5: Affine coordinate rings and finitely-generated reduced $k$-algebras

Show that a $k$-algebra $B$ is isomorphic to the affine coordinate ring of some algebraic set in $\mathbb A^n$, for some $n$, if and only if $B$ is a finitely generated $k$-algebra with no nilpotent elements.

There are two directions to prove here. First, if $A=k[x_1,\dots,x_n]$ and $B=A/I(Y)$ is the affine coordinate ring of any algebraic set $Y$, we must find a canonical realization of $B$ as a $k$-algebra. To do this, we consider the composition $$k\hookrightarrow A\twoheadrightarrow A/I(Y)$$ where the first map takes elements of $k$ to their associated constant polynomials and the second map is the canonical quotient map. The first of these maps is finitely generated since it is just the polynomial algebra, and the second map is finitely generated since $A$ is noetherian (Hilbert's basis theorem). Thus the composition is also finitely generated. Further, since $I(Y)$ is a radical ideal the quotient $A/I(Y)$ is reduced.

Conversely, if $k\to B$ is any finitely generated $k$-algebra then we can realize $B$ as the quotient of some polynomial ring $k[x_1,\dots,x_n]$ by an ideal $\mathfrak a$. Since $A/\mathfrak a=B$ is finitely generated by assumption, the ideal $\mathfrak a$ is also finitely generated. It is a radical ideal since $B$ is a reduced ring. Set $Y=Z(\mathfrak a)$, then $$A/I(Y) = A/\sqrt{\mathfrak a}=A/\mathfrak a=B$$ as desired.