Let $R$ be a domain with $Frac(R)=K$. $R$ is called valuation ring of field $K$ if $\forall a,b\in R$ either $aR\subset bR$ or $bR\subset aR$.
Let $R$ be a valuation ring of field $K$. Let $p$ be maximal ideal of $R$. Denote $R^\star$ as a valuation ring of field $\frac{R}{p}$. Let $R'=\{x\in R\vert x(\mod p)\in R^\star\}$. Then $R'$ is a valuation ring of $K$.
$\textbf{Q:}$ What is the application of above statement? It seems this incorporates both residue field and elements of $R$. I could not see any application to number ring settings.
This construction corresponds to the composition of valuations: the valuation ring $R$ corresponds to a valuation $v: K^\times\to \Gamma$ on the fraction field, and $R^\star$ to a valuation $w:\kappa^\times \to G$ where $\kappa=R/p$ is the residue field.
Then $R'$ corresponds by definition to the valuation $w\circ v$ on $K$.
It is often use for induction processes, if for instance the value group is $\mathbb{Z}^n$ (you can think of $k((X_1,\dots,X_n))$ for instance), then it can be decomposed as a composition of $n$ valuations with value group $\mathbb{Z}$, which are easier to work with.