Calculation of length of modules

Question

I am trying to calculate the length of two modules: $k[x,y]_{(x,y)}/(y-x^2,y)$ and $k[x,y]_{(x,y)}/(y-x^2,x)$. The claim is that the former has length 2 and the latter has length 1. But I am not sure why this is true: the chain of ideals of $k[x,y]_{(x,y)}$ containing $(y-x^2,y)$ can be $(y-x^2,y) \subset (x,y) \subset k[x,y]_{(x,y)}$, so it should have length 3. Similarly I have $(y-x^2,x) \subset (x,y) \subset k[x,y]_{(x,y)}$ so the length of $k[x,y]_{(x,y)}/(y-x^2,x)$ should also be 3. Can somebody please point out what's wrong with my method?

Answer 1

The first filtration is true, but the length is 2 because $(y-x^2,y)=0$ in the ring.

The second filtration is not true, since $(y-x^2,x)=(x,y)$, hence the length is 1 by the same reason.