Let R be a noetherian ring and V a valuation ring with maximal ideal $\mathfrak{m}_V$. Does every morphism of rings $\varphi: R \rightarrow V$ factor through a discrete valuation ring?

One may consider the homomorphic image $S$ of $\varphi$ in V. This is a noetherian subring and we may even assume it to be local by localizing at $\mathfrak{m}_S :=\mathfrak{m}_V \cap S$. Then we can find by a standard procedure a discrete valuation ring $T$ dominating $S$ in $L= Frac(S)$:

We can consider the blowup $\tilde{S}$ of $S$ in its closed point $\mathfrak{m}_S$ and pick a generic point $\mathfrak{n}$ of an irreducible component of the exceptional divisor.

Then $\mathcal{O}_{\tilde{S},\mathfrak{n}}$ is a 1-dimensional local ring with field of fractions $L$ which dominates $S$.

Normalizing $\mathcal{O}_{\tilde{S},\mathfrak{n}}$ in $L$ yields the desired DVR $T$.

I guess the question is if we may choose $\mathfrak{n}$ above in a way such that $T \subset V \subset Frac(V)$. I would be also happy with extending $V$ such that this statement holds.

Is a statement like this known to be true or false?

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