Let $R \subseteq \mathbb C[X,Y]$ be the subring of all polynomials $f \in \mathbb C[X,Y]$ that can be written as $f = g(X)+X ·h(X,Y)$
a. Let $I \subset R$ be the kernel of the evaluation map $R \rightarrow C$ given by$ f \mapsto f(0,0)$. Show that the ideal $ I$ is not finitely generated.
b. Conclude that in general a submodule of a finitely generated module need not be finitely generated
I previously asked this question here.
I got an aswer there, but it looks pretty convoluted to me, there are several things I am not convinced about, and with an answer already there no one else is helping. this question is just to get my solution checked and possibly another suggestion
I have a proposal, but I need someone to tell me if it is correct/ help me fill in the details I've been browsing around and looks like the standard procedure is to get a contradiction in the equation that expresses an element of the set that I want to prove not being finitely generated as generated by the elements of a finite generator set. I am basically drawing inspiration from this question.
My proposal:
Let $ev(f):=f(0,0)$ denote the evaluation map.
The first step is to realized that the polynomials in $R$ can be written as $ f= g(0)+X\tilde h(X,Y)$, while those in the ideal as $ f=X\tilde h(X,Y)$.
$I= \text{Ker (ev)}=\{f \in R: ev(f)=f(0,0)=0\}=\{ f(x,y)=g(x)+x·h(x,y): g(0)=0\}=X\mathbb C[X;Y]$
I supposed the question is asking about the ideal being generated as $R$-module? I am not sure if this question is compatible with regarding a ring with unity as we do in my course, because that ideal does not contain the unity.
The ideal can be seen as generated by monomials of the form $f_k=xy^k$, so $I=\langle x, xy, xy^2, \dots \rangle$.
By absurd suppose that $I$ is finitely generated. then $\exists$ polynomials $f_1, f_2,...,f_n \in I$ such that $\forall f \in I, f=\sum_{i=1}^{n} r_if_i$ with $r_i \in R$.
Take the highest $y$ power appearing in each generator in $\{f_1, \dots, f_m\}$ and call this number $N$, then $xy^{N+1}\not\in I$.
If this is wrong, why? Can you suggest how to fix it?