 Solution

An affine variety in ${\mathbb{A}}_k^n$ is the common zero set of a collection of polynomials in $k[x_1,\dots ,x_n]$, with its Zariski topology. We use the convention of this question that such a variety need not be irreducible. Its vanishing ideal and coordinate ring areThe quotient identifies polynomials that give the same function on $X$, so its elements are exactly polynomial functions on $X$.

For affine embeddings $X\subseteq {\mathbb{A}}^n$ and $Y\subseteq {\mathbb{A}}^m$, a morphism of affine varieties is a map $F=(f_1,\dots ,f_m)$ with $f_i\in A(X)$ and image in $Y$. Pullback of polynomial functions defines the contravariant homomorphismIt is well defined because any polynomial vanishing on $Y$ pulls back to zero on $X$, and it preserves addition, multiplication, one, and the constants in $k$.