Exercise 13 (b) of Chapter III in Lang's Algebra is as follows.

Let $M$ be a finitely generated projective module over the Dedekind ring $\mathfrak{o}$. Then there exists free modules $F$ and $F^\prime$ such that $F \supset M \supset F^\prime$ were $F$ and $F^\prime$ have the same rank $n$. Prove that there exists a basis $\{e_1, \cdots, e_n \}$ of $F$ and ideals $\mathfrak{a}_1, \cdots, \mathfrak{a}_n$ such that $M = \mathfrak{a_1} e_1 + \cdots + \mathfrak{a_n} e_n$, or in other words, $M \approx \oplus \; \mathfrak{a}_i$.

I understand the proof of the structure theorem for finitely generated modules over Dedekind domains, but I do not see how to prove the existence of a basis of $F$ if we follow the same kind of arguments.


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