Embedding a ℤ-finite domain into a complete DVR

Let $R$ be a commutative ring which is an integral domain of characteristic zero and finitely generated as a $\mathbb{Z}$-module, let $p$ be a prime number, let $F$ be a field of characteristic $p$, and let $\pi \colon R \to F$ be a ring homomorphism. The assertion is the existence of the following data: a commutative ring $\mathcal{O}$ which is an integral domain, a discrete valuation ring, adically complete with respect to its maximal ideal $\mathfrak{m}_{\mathcal{O}}$, with finite residue field $\mathcal{O}/\mathfrak{m}_{\mathcal{O}}$ and of characteristic zero; a ring homomorphism $\psi \colon R \to \mathcal{O}$; a field $F'$ equipped with an $F$-algebra structure; and a ring homomorphism $\iota \colon \mathcal{O}/\mathfrak{m}_{\mathcal{O}} \to F'$, such that $\psi$ is injective, the preimage $\psi^{-1}(\mathfrak{m}_{\mathcal{O}})$ equals $\ker \pi$, the image of $p$ in $\mathcal{O}$ lies in $\mathfrak{m}_{\mathcal{O}}$, and for every $x \in R$ one has $\iota(\psi(x) \bmod \mathfrak{m}_{\mathcal{O}}) = \pi(x)$ in $F'$, the right-hand side being taken via the structure map $F \to F'$. Thus $\pi$ is recovered, after the extension $F \to F'$, from reduction of $\psi$ modulo $\mathfrak{m}_{\mathcal{O}}$.

This is the standard passage from an abstract $\mathbb{Z}$-finite coefficient domain with a characteristic-$p$ character to a $p$-adic coefficient ring: the fraction field of $R$ is a number field, $\ker \pi$ is a prime above $p$, and $\mathcal{O}$ may be taken to be the completion of the ring of integers at a prime lying over it. It supplies the complete discrete valuation coefficient ring needed in WeierstrassCurve.isModularModelOfLevel_div_of_isGoodPrimeFor_of_dvd_of_not_sq_dvd, where Hecke eigenvalues living in a $\mathbb{Z}$-finite ring must be compared with a mod-$p$ system.