Rings of polynomials whose (partial) derivatives vanish are Noetherian? I have the following two questions on rings of polynomials. They seemed similar enough that I thought I'd go ahead and group them here as opposed to making separate listings for them. The questions are:

($1$) Determine whether or not the ring of polynomials in $z$ whose first $k$ derivatives vanish at the origin for some fixed positive integer $k$ is Noetherian.
($2$) Determine whether or not the ring of polynomials in $z$ and $w$ whose partial derivatives with respect to $w$ vanish for $z=0$ is Noetherian.

Note that $z$ and $w$ are complex variables, that is, the polynomials are over $\Bbb C$.
My work: It seemed like I could view each of these as subrings of $\Bbb C[z]$ and $\Bbb C[z,w]$, and since both of these are Noetherian, I wanted to conclude that each of the above were Noetherian, but I realized it isn't necessarily the case that a subring of a Noetherian ring is Noetherian. Can anyone offer some help for me? Thanks!
 A: (1) $R=\mathbb C+X^{k+1}\mathbb C[X]$, and this is a subring of $\mathbb C[X]$. Furthermore, the extension $R\subset \mathbb C[X]$ is finite and $\mathbb C[X]$ is noetherian, hence $R$ is noetherian. (An elementary approach: set $S=\mathbb C[X^{k+1}]$. Then $R$ is a finitely generated $S$-module (generated by $1,X^{k+2},\dots,X^{2k+1}$), and therefore $R$ is a noetherian $S$-module. It follows that $R$ is a noetherian ring.) 
(2) $R=\mathbb C[X]+X\mathbb C[X,Y]$. This ring is not noetherian since the ideal $I=X\mathbb C[X,Y]$ is not finitely generated. Assume by contrary that $I$ is finitely generated and let $Xf_1(X,Y),\dots,Xf_n(X,Y)$ be a system of generators. Then every element of $I$, say $Xg(X,Y)$, can be written as follows: $$Xg(X,Y)=Xf_1(X,Y)[a_1(X)+Xb_1(X,Y)]+\cdots+Xf_n(X,Y)[a_n(X)+Xb_n(X,Y)].$$ Then $$g(X,Y)=f_1(X,Y)[a_1(X)+Xb_1(X,Y)]+\cdots+f_n(X,Y)[a_n(X)+Xb_n(X,Y)].$$ Now we send $X$ to $0$ and get $$g(0,Y)=f_1(0,Y)a_1(0)+\cdots+f_n(0,Y)a_n(0).$$ This shows that every polynomial of $\mathbb C[Y]$ can be written as a linear combination of $f_1(0,Y),\dots,f_n(0,Y)$ with complex coefficients, a contradiction.
A: In question (1) the  ring obtained for $k=1$ is $S=\mathbb C+z^2\mathbb C[z]$.
   That ring is noetherian because it is isomorphic to $\mathbb C[X,Y]/(Y^2-X^3)$ : can you show that?
