Let $R$ be a ring and $M$ an $R$-module.
If we suppose $M$ finitely generated then (following Matsumura) if we write $M=Rm_1+\cdots+Rm_n$ we have:
$p\in\operatorname{Supp}M$ if and only if $M_p\neq0$ if and only if there exists an $i$ such that $m_i\neq0$ in $M_p$ if and only if there exists an $i$ such that $\operatorname{Ann}m_i\subset p$ if and only if $\operatorname{ann}M=\bigcap_{i=1}^n\operatorname{Ann}m_i\subset p$. And so $\operatorname{Supp}M=V(\operatorname{Ann}M)$.
If $M$ is not finitely generated where does this proof fail?
It seems to me that if we write $M=\langle m_i\rangle_{i\in I}$ nothing will change.
And if this proof fails could you give me an example of a module such that $\operatorname{Supp}M\neq V(\operatorname{Ann}M)$?