Probably this is just a misunderstanding, but it's making me waste a lot of my time. I'm studying commutative algebra through Kemper's book. There is the following statement in chapter 12 (dimension theory).
But if we suppose $\mathfrak{q}\subsetneq\mathfrak{m}$, then we get the chain of prime ideals $\{0\}/\mathfrak{q}\subsetneq\mathfrak{m}/\mathfrak{q}$ in $R/\mathfrak{q}$, therefore $\dim(R/\mathfrak{q})=1$. This is so straightforward so I can't get the error, what is wrong with this reasoning?
Thanks for helping.
PS: I already know that $\mathfrak{m}/\mathfrak{q}=nil(R/\mathfrak{q})$, but this didn't help.
$\mathfrak{q}$ is not assumed prime, so that $R/\mathfrak{q}$ is not a domain. In fact, $\sqrt{\mathfrak{q}} = \mathfrak{m}$ implies that the intersection of all prime ideals containing $\mathfrak{q}$ is $\mathfrak{m}$. This can only happen if $\mathfrak{m}$ is the only prime ideal containing $\mathfrak{q}$; $\mathfrak{q}$ included.