Suppose that $R$ is a commutative Noetherian local ring, $M$ a finitely generated $R$-module and $\mathfrak{p}\in \mathrm{Supp}_R(M)$. Then I am little confused with the following equality between Krull dimensions: \begin{equation} \dim_R M/\mathfrak{p}M=\dim_{R/\mathfrak{p}} M/\mathfrak{p}M.\end{equation}
If the above equality holds, then have the following two possibilities:
(1) $\dim_R M/\mathfrak{p}M=\dim_{R/\mathfrak{p}} M/\mathfrak{p}M\leq \dim_{R/\mathfrak{p}} R/\mathfrak{p}=\dim_R R/\mathfrak{p}$.
(2) As $\mathfrak{p}\in \mathrm{Supp}_R(M/\mathfrak{p}M)=\mathrm{Supp}_R(M)\cap V(\mathfrak{p})$, it follows that $\dim_R M/\mathfrak{p}M\geq \dim_R R/\mathfrak{p}$, which is a contradiction of above statement (1).
Can anyone guide me where am I wrong? I think my first equation is not true. But I do not have any example or argument.