Vanishing of a finitely generated module near a prime ($M_P=0⟺{\mathrm{Ann}}_R(M)⊈P$)

For a finitely generated module $M=(m_1,\dots ,m_t)$ and a prime ideal $P$, if $M_P=0$, choose $s_i\notin P$ annihilating $m_i$. Their product $f\notin P$ annihilates all of $M$. Conversely such an $f$ becomes a unit in $R_P$, forcing $M_P=0$. ThusThis identifies the support of a module $M$ with the Zariski-closed set $V({\mathrm{Ann}}_R(M))$. For $M=0$, use $f=1$. No Noetherian assumption on $R$ is required.