Let $A$ be a Dedekind domain, meaning a commutative Noetherian integrally closed domain of Krull dimension $1$, and let $K=\operatorname{Frac}(A)$. Let $\mathcal I(A)$ be the abelian group, under multiplication, of nonzero fractional ideals of $A$, meaning nonzero $A$-submodules $I\subset K$ for which there exists $d\in A\setminus\{0\}$ with $dI\subset A$. Let $\mathcal P(A)=\{aA:a\in K^\times\}$ be the subgroup of principal fractional ideals, and define $\operatorname{Cl}(A)=\mathcal I(A)/\mathcal P(A)$. Let $\operatorname{Pic}(A)$ be the group of isomorphism classes of invertible $A$-modules, equivalently finitely generated projective $A$-modules of constant rank $1$, with group law induced by [tensor product](/page/Tensor%20Product) over $A$. Then the assignment sending a nonzero fractional ideal $I\subset K$ to its isomorphism class $[I]$ as an invertible $A$-module induces a group isomorphism $\operatorname{Cl}(A)\cong \operatorname{Pic}(A)$.