Many introductory texts on algebraic geometry set up some sort of algebra-geometry dictionary in which radical ideals correspond to varieties, and so on. I am wondering if there is a geometric way to think about a subalgebra in a polynomial ring? For instance for invariant rings one may develop some intuition, but what about infinitely generated subalgebras? A simple example would be the infinitely generated monomial algebra
$k[x,xy, xy^2, xy^3, \dots]$ as a subalgebra in $k[x,y]$ for some field $k$.



What is the geometric object corresponding to a subalgebra in a polynomial ring
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 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.


