Let $A$ be a Dedekind domain with fraction field $F$. Let $A^\times$ and $F^\times$ denote the multiplicative groups of units of $A$ and nonzero elements of $F$, respectively. Let $\operatorname{Div}(A)$ denote the free abelian group on the nonzero prime ideals of $A$. Let $\operatorname{Cl}(A)$ denote the ideal class group of $A$, defined as the group of nonzero fractional ideals of $A$ modulo the subgroup of principal fractional ideals. Let

\begin{align*} \operatorname{div}:F^\times \to \operatorname{Div}(A) \end{align*}

be the principal divisor homomorphism defined by

\begin{align*} \operatorname{div}(x)=\sum_{\mathfrak p \ne 0} v_{\mathfrak p}(x)[\mathfrak p], \end{align*}

where $v_{\mathfrak p}(x)$ is the exponent of $\mathfrak p$ in the factorization of the principal fractional ideal $xA$. Then the sequence

\begin{align*} A^\times \longrightarrow F^\times \xrightarrow{\operatorname{div}} \operatorname{Div}(A) \longrightarrow \operatorname{Cl}(A) \longrightarrow 0 \end{align*}

is exact, where the first map is the inclusion and the map $\operatorname{Div}(A)\to\operatorname{Cl}(A)$ sends $\sum_{\mathfrak p} n_{\mathfrak p}[\mathfrak p]$ to the ideal class of $\prod_{\mathfrak p}\mathfrak p^{n_{\mathfrak p}}$.