Let $R$ be a Dedekind Ring and $K = \operatorname{Quot}(R)$. Let $\mathcal{I}_K$ be the ideal group and $C \ell_K$ the ideal class group.
In a lecture in algebraic number theory, our professor remarked that
$$1 \to R^{\times} \to R(X)^{\times} \to \bigoplus_{p\notin X}K^{\times} / R^{\times}_p \to \mathcal{C} \ell(R) \to \mathcal{C} \ell(R(X)) \to 1$$ Where ,
R(X)= {$\frac{f}{g} : f,g \in R, g \not\equiv 0$mod $p$ for $p\in X$}
$\mathcal{C} \ell(R)$ and $\mathcal{C} \ell(R(X))$ are ideal class groups.
Prove this sequence is exact. (I am having trouble understanding the proof of exactness at $\bigoplus_{p\notin X}K^{\times} / R^{\times}_p $ part.)
(Source :Algebraic Number Theory by Neukirch, Proposition 11.6) Is there any other book, where I can find the proof of this exact sequence ?