Proposition 24

Let $A$ be a Dedekind domain and $M$ a finitely generated torsion-free $A$-module & rank $n \geq 1$. Then there exists an ideal $b \neq 0$ of $A$ such that $M$ is isomorphic to the direct sum of the modules $A^{n-1}$ and $b$. Moreover, the class of the ideal $b$ is determined uniquely by this condition.

Theorem 6 of no. 9 shows that there exists a free submodule $L$ of $M$ such that $M/L$ is isomorphic to an ideal $a$ of $A$. If $a = 0$, we take $b = A$. Otherwise, $a$ is of rank 1, hence $L = A^{n-1}$ and $a$ is a projective module (Proposition 22); $M$ is therefore isomorphic to the direct sum of $L$ and $a$ (Algebra, Chapter II, § 2, no. 2, Proposition 4), which proves the first part of the proposition. Moreover, it follows from no. 7, Proposition 16 (i), (iv) and (v) that $c(M) = c(b)$ whence the uniqueness of the class of $b$.

Remarks (1) Propositions 23 and 24 and the Corollary to Proposition 22 determine completely the structure of finitely generated $A$-modules. Proposition 24 shows that a torsion-free finitely generated $A$-module is determined up to isomorphism by its rank and the divisor class which is attached to it. (2) It can be shown that over a Dedekind domain a projective module which is not finitely generated is necessarily free (Exercise 21) and that every submodule of a projective module is projective (Exercise 20).