Here is the problem I am stuck on: Fix a field $K$ and consider the subring $A \leq K[x,y]$ generated by $K \cup \{x,xy,\dots,\}$. Show that $A$ is not Noetherian.
I figure that taking ideals $I_n = (x,xy,\dots,xy^n)$ should give an infinite strictly ascending chain, which would establish that $A$ is not Noetherian, but I cannot figure out how to show that each $I_n$ is a proper subset of $I_{n+1}$. Any help here?
Well, $xy^n$ is in $I_n$, but not in $I_{n-1}$. Indeed, you can show that the only monomials in $I_n$ that are not divisible by $x^2$ are scalar multiples of the ones generating $I_n$: we have:
$I_n = \{ \sum_{p = 0}^n f_i x y^{p}: f_1, \ldots, f_n \in A\}$
Thus, the monomials in $I_n$ are just $f x y^{p}$ for $0 \leq p \leq n$ and some monomial $f \in A$. If $f \in K$, then we get scalar multiples of generators, and if $f \not \in K$, then $f$ is divisible by $x$ (as all nonscalar monomials in $A$ are), so we get monomials divisible by $x^2$.
