Projections between two affine algebraic sets

Question

What are the two projections $\mathbb{A}^2\to\mathbb{A}^1$, that exhibits $\mathbb{A}^2$ as a product $\mathbb{A}^1\times\mathbb{A}^1$ in the category of affine algebraic sets?

Answer 1

$\operatorname{Spec}A \times \operatorname{Spec}B$ is $\operatorname{Spec} (A \otimes B),$ and accordingly the projections are induced by canonical inclusions in the tensor product. In particular, projections $\mathbb{A}_k^2\to\mathbb{A}_k^1$ come from $k[x] \hookrightarrow k[x,y]$ and $k[y] \hookrightarrow k[x,y].$