Maybe this question is trivial : most of the algebras I encounter (issued from invariant theory !) are of finite type. Does someone have a 'simple' example of an algebra/subalgebra of infinite type. In fact, I don't see immediately how to show that some algebra is of infinite type.
For instance, if we take the subalgebra $$\mathbb{R}[x^{k}+k,\quad k\geq 2]$$ it seems to me to be of infinite type...
Same for $$\mathbb{R}[x^{k_1}y^{k_2},k_1\geq 1,k_2\geq 2]$$
Is it obvious ?
Any finite type $K$-algebra is isomorphic to $K[X_1,\ldots,X_n]/I$, for some $n\geq 0$ and some ideal $I$. In particular, a finite type $K$-algebra is a Noetherian ring.
Hence any non Noetherian ring containing a field $K$ as a subring will give you a counterexample. The simplest one is $K[X_m, m\geq 0]$.
If you want an example of a finite type algebra having a subalgebra not of finite type, you can take $K[X,Y]$ and $K[XY^k,k\geq 0]$ , since the second one is not Noetherian (see How to prove that $k[x, xy, xy^2, \dotsc]$ is not noetherian?)
