I am currently working through Algebraic Number Fields by Janusz and towards the bottom of page 19 he leaves an exercise to the reader. The exercises states
Let $S$ be a multiplicative set in the Dedekind Ring $R$. Show that the inclusion map $R\rightarrow R_S$ induces a group epimorphism $\mathbf{C}(R)\rightarrow\mathbf{C}(R_S)$.
Here $\mathbf{C}(R)$ and $\mathbf{C}(R_S)$ denote the class groups of $R$ and $R_S$, respectively. Janusz defines the class group by first defining $\mathbf{I}(R)$ to be the set of fractional ideals of $R$ and $\mathbf{P}(R)$ to be the set the principal fractional ideals of $R$. Then $$\mathbf{C}(R)=\mathbf{I}(R)/\mathbf{P}(R).$$ I understand that for any fractional ideal $\mathfrak{M}_S$ of $R_S$ we can find a fractional ideal $\mathfrak{M}=\mathfrak{M}_S\cap R$ of $R$, and for any fractional ideal $\mathfrak{N}$ of $R$ we have that $\mathfrak{N}R_S$ is a fractional ideal of $R_S$.
(Note: Multiplication of fractional ideals is defined just like multiplication of ideals)
Is this the correspondence we are using to define the induced epimorphism from $\mathbf{C}(R)$ to $\mathbf{C}(R_S)$? I suppose I do not understand what he means by an epimorphism induced by the inclusion map in this particular case. Any insight and help is appreciated.
You start by defining a map $\phi\colon I(R)\to I(R_S)$.
If you know tensor products, then this is quite easily just the map $\phi(J)= J \otimes_R R_S$. Otherwise you can also define it as $\phi(J) = \{x \in \operatorname{Frac} R: \exists s \in S, sx \in J\}$.
You then need to verify that $\phi$ sends a principal ideal to a principal ideal, therefore inducing a map $f\colon C(R) \to C(R_S)$.
Finally you need to verify that $f$ is an epimorphism. This is more or less carried out in your question (the part starting with "I understand that ...").