Length of the primary component of $(xy, y^2)$ at the origin is $1$. As the question title suggests, how do I see that the length of the primary component of $(xy, y^2)$ at the origin is $1$?

We want to calculate the length of the longest ideal of finite length in the ring$$R_\mathfrak{p}/IR_\mathfrak{p}.$$For us, $\mathfrak{p} = \langle x, y\rangle$ and $I = \langle y^2, xy\rangle$. Since $R_\mathfrak{p}$ is a local ring, the obvious choice of an idea of maximal length is $\langle x, y\rangle$. This ideal is a vector space of dimension one over $k$. So the length is obviously one.
