Suppose $A$ is a (possibly noetherian) ring and M a (possibly finitely generated) $A$-module. Consider for $m \in M$ the set $$\{m = 0\} := \{\mathfrak p \subset A \text{ prime ideal} \,|\, m_{\mathfrak p} \in \mathfrak p M_{\mathfrak p} \} \subset \operatorname{Spec}(A).$$ Is $\{m = 0\}$ a closed subset of $\operatorname{Spec} A$? If so, does it have a "better" canonical subscheme structure than the reduced one?
I realized that I should at least suppose that $\operatorname{Ann}(m) = 0$, because for $\mathfrak p \notin V(\operatorname{Ann}(m))$, we have $m_{\mathfrak p} = 0$. So for example if $A = k[x,y]$, then $$m = \overline y \in k[x,y] / (x)$$ vanishes exactly on $\mathbb A^2 \setminus \{x = 0, y \neq 0\}$.