Let $f:(R,\mathfrak m) \to (S,\mathfrak n)$ be an integral local homomorphism. Let $\mathfrak p$ be a prime ideal of $R$ not equal to $\mathfrak m$. I want to know if one can claim $\dim S/f(\mathfrak p)S\neq0 $.
This is true when $(f(\mathfrak p)S)^c=\mathfrak p$, but in general?
Thanks.