Example

If $A = \mathbf{Z}$, the divisor group of $A$ is identified with the multiplicative group $\mathbf{Q}_+^*$ of rational numbers > 0. For every finite commutative group T, $\chi(T)$ is the order of T; the above corollary shows that the order of the group $\operatorname{Coker}(f)$ is equal to the absolute value of $\det(U)$ (cf. Algebra, Chapter VII, § 4, no. 7, Corollary 3 to Theorem 3).