motivation for definition of class group by modding out principal ideal group

Question

Given $R$ dedekind domain, it is clear that every ideal $I\subset R$ has $I^{-1}$. Furthermore, any principal ideal is clearly invertible. In $R$, all ideals forms an abelian group G. And principal ideals form a subgroup $H$ of this abelian group G. One define the class group by $G/H$. What is motivation for modding out this $H$? It looks like I am ignoring the blob generated by $(0)$ ideal. I found a pure number theoretical answer ( Motivation behind the definition of ideal class group). Can someone kindly provide a geometrical interpretation?

Answer 1

Geometrically, a fractional ideal of a domain $R$ is a line bundle $L$ on $X=\operatorname{Spec}X$ together with a fixed embedding $L\to\mathcal K_X$ into the sheaf of rational functions on $X$. Modding out the principal ideals is the same, under this correspondence, to forgetting the embedding into $\mathcal K_X$

Answer 2

Since a Dedekind domain is a UFD iff it is a PID, the class group measures how much unique factorization fails in $R$ by measuring how far the ideals are from being principal.