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!