I am trying to prove that, for a given field $F$, the subring $$R:=\{p(X,Y)=\sum c_{ij}X^iY^j \in F[X,Y] : c_{0j}=c_{j0}=0 \text{ whenever } j>0\}$$ of $F[X,Y]$ is not Noetherian.
I think I have done it, but I'm new to this stuff so I would really like someone to check it for me.
Here is my solution:
My idea is to exhibit an ascending chain of ideals which does not terminate (this seems to be the easiest way to proceed out of the 3 possible definitions of Noetherian that I know).
The general form of an element in $R$ is $$c + XYf(X,Y)$$ where $c \in F$ and $f(X,Y)\in F[X,Y]$.
I claim that $(XY)\subsetneq (XY,X^2Y) \subsetneq (XY,X^2Y,X^3Y)\subsetneq\cdots$ is a non-terminating chain.
The general form of an element of the ideal $(XY,\ldots,X^nY)$ is $$c_1XY+\cdots+c_nX^nY+Y^2[X^2f_1(X,Y)+\cdots+X^{n+1}f_n(X,Y)]$$ where $c_i \in F$ and $f_i\in F[X,Y]$ and this clearly does not have a $X^{n+1}Y$ term.
Does this proof work?
Thank you!
