I have the following situation: $R\subset T \subset S$ are rings, where I know that $T$ is integral over $R$ and $S$ is integral over $R$, does it imply that $S$ is integral over $T$?
I am trying to prove that $S = k[y_1,\ldots, y_d]$ is integral over $T = k[u, z_2,\ldots, z_{d}]$ where $u$ is a non-zero polynomial over $k$, and $T$ is integral over $R = k[z_2,\ldots, z_{n-1}]$.
All this is to understand following line from the proof of $\dim(k[y_1,\ldots, y_d])=d$ given in wikipedia.
We apply the noether normalization and get $T=k[u,z_{2},\ldots ,z_{d}]$ (in the normalization process, we're free to choose the first variable) such that $S=k[y_1,\ldots, y_d]$ is integral over $T$.