ag.algebraic geometry – Determine vanishing of a section from its zero loci

Let $mathcal{F}$ be a coherent sheaf on a complex algebraic variety $X$. In this paper (section 5), it is proved that the zero loci of a section $s$, denoted by $Z(mathcal{F}, s)$, is constructible.

Question: is it true that $s=0$ if and only if $Z(mathcal{F}, s)=X$? If so, how to prove this?

Since everything is local, we can assume that $X=text{Spec} R$, and then $mathcal{F}$ corresponds to a finitely generated $R$-module $M$. For any prime ideal $mathfrak{p}subset R$, let $kappa(mathfrak{p})$ be the residue field at $mathfrak{p}$. Recall the zero loci of $sin M$ is defined as
begin{align*}
Z(M, s)={mathfrak{p}: overline{s}in Motimes_R kappa(mathfrak{p}) text{ is }0}.
end{align*}