How can I prove that the Krull dimension of $R[X]/(f(X))$, for $R$ a finitely generated noetherian integral domain and $f(X)$ monic, is equal to the Krull dimension of $R$?
I don't even know where to start, since even to use Noether normalization I would need $(f(X))$ to be prime, right? Any help would be great.
There is no need to assume $R$ domain or noetherian.
We have that $R\subset R[X]/(f)$ is an integral extension, so $\dim R[X]/(f)=\dim R$. (Of course, we suppose $\deg f\ge 1$.)