Let $(X,\mathcal{O}_X)$ be a locally ringed space and let $\mathcal{F}$ be an $\mathcal{O}_X$-module. For a section $s \in \mathcal{F}(X)$ and a point $x$, we say $s(x)=0$ if the stalk $s_x$ is zero modulo the (unique) maximal ideal of $\mathcal{O}_{X,x}$.
Is it true in general that $\{x \in X | s(x) = 0 \}$ is closed?
I think I proved it for locally free $\mathcal{O}_X$-modules: if one of the components of a section doesn't vanish, then this component is invertible (in particular non-zero) in an open neighbourhoud. But I'm not able to extend this proof to $\mathcal{O}_X$-modules in general.
When $X=\mathrm{Spec}(\mathbb{Z})$ and $\mathcal{F}$ is the quasi-coherent sheaf associated to $\mathbb{Z}/p \mathbb{Z} = \langle s \rangle$ for a prime $p$, prove that $\{x \in X : s(x)=0\} = X \setminus \{(p)\}$, which is not closed.