Ideal class group of an extension of rings and Picard group

For any extension of commutative rings $A\subseteq B$, by using invertible ideals, we first define an Abelian group $\text{Cl}(A,B)$, that we call the ideal class group of this extension. Then we study the main properties of this group. Among them, we prove that the group $\text{Cl}(A,B)$ is indeed the kernel of the natural group morphism $\text{Pic}(A)\to \text{Pic}(B)$ which is given by $L\mapsto L{\otimes }_{A}B$. Then we show that both the classical ideal class group and, surprisingly, the Picard group are special cases of this structure. Next, we prove that …