Because geometry and algebra are connected by the contavariant $\textrm{Spec}$ functor, a subalgebra corresponds to the image of a dominant morphism. So subalgebras of $k[x_1,..,x_n]$ are affine schemes with dominant maps from $\mathbb A^n_k$.

Sometimes, like in your example, this is in addition a surjective morphism, so it looks like the quotient of $\mathbb A^n$ by some equivalence relation. But sometimes, the simplest example being $k[x,xy] \subset k[x,y]$, it fails to be surjective. Here there is a map from the whole plane to the plane whose image is the plane minus a line plus a point.

Since the image is always dense, we can geometrically think of it as the quotient of $\mathbb A^n_k$ by an equivalence relation followed by a completion or partial compactification.

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Because geometry and algebra are connected by the contavariant $\textrm{Spec}$ functor, a subalgebra corresponds to the image of a dominant morphism. So subalgebras of $k[x_1,..,x_n]$ are affine schemes with dominant maps from $\mathbb A^n_k$.

Sometimes, like in your example, this is in addition a surjective, so it looks like the quotient of $\mathbb A^n$ by some equivalence relation. But sometimes, the simplest example being $k[x,xy] \subset k[x,y]$, it fails to be surjective. Here there is a map from the whole plane to the plane minus a line plus a point.

Since the image is always dense, we can geometrically think of it as the quotient of $\mathbb A^n_k$ by an equivalence relation followed by a completion or partial compactification.