Let $X$ be a noetherian, locally factorial scheme. Then the homomorphism \begin{equation*} \mathrm{cyc} : \mathrm{Div}(X)\longrightarrow\mathrm{Z}^{1}(X) \end{equation*} from Cartier divisors to prime Weil divisors is surjective, [Görtz&Wedhorn, p.307].
I am not sure with the trivial part of the above proof, namely $\mathrm{cyc}(d(Z)) = Z$. I don't see, why for the generic point $z\in X$ of the prime Weil divisor $Z\subset X$ the maximal ideal of $\mathcal{O}_{X,z}$ is generated by $f_{z}\in\mathcal{O}_{X,z}$.