Let $M$ be a finitely generated torsion-free module over a Prüfer domain $R$ with field of fractions $k$ . We show that $M$ is isomorphic to a direct sum of finitely generated ideals in $R$ .

We shall write $k\otimes M$ for the vector space over $k$ generated by $M$ . This is just the localization of $M$ at $R\setminus\{0\}$ and, as $M$ is torsion-free, the natural map $M\rightarrow k\otimes M$ is one-to-one and we can regard $M$ as a subset of $k\otimes M$ .

As $M$ is finitely generated, the vector space $k\otimes M$ will finite dimensional, and we use induction on its dimension $n$ . Supposing that $n>0$ , choose any basis $e_1,\ldots,e_n$ and define the linear map $f\colon k\otimes M\rightarrow k$ by projection onto the first component, \begin{equation*} f(x_1e_1+\cdots+x_ne_n)=x_1. \end{equation*}Restricting to $M$ , this gives a nonzero map $M\rightarrow k$ . Furthermore, as $M$ is finitely generated, $f(M)$ will be a finitely generated fractional ideal in $k$ . Choosing any nonzero $c\in R$ such that $\mathfrak{a}\equiv cf(M)\subseteq R$ , \begin{equation*} g\colon M\rightarrow\mathfrak{a},\ g(u)=cf(u) \end{equation*}defines a homorphism from $M$ onto the nonzero and finitely generated ideal $\mathfrak{a}$ . As $R$ is Prüfer and invertible ideals are projective, $g$ has a right-inverse $h\colon\mathfrak{a}\rightarrow M$ . Then $h$ has the left-inverse $g$ and is one-to-one, so defines an isomorphism between $\mathfrak{a}$ and its image. We decompose $M$ as the direct sum of the kernel of $g$ and the image of $h$ , \begin{equation*} M = \operatorname{ker}(g)\oplus \operatorname{Im}(h)\cong\operatorname{ker}(g)\oplus \mathfrak{a}. \end{equation*}Projection from the finitely generated module $M$ onto $\operatorname{ker}(g)$ shows that it is finitely generated and, \begin{equation*} \operatorname{dim}(k\otimes\operatorname{ker}(g))=\operatorname{dim}(k\otimes M)-\operatorname{dim}(k\otimes\mathfrak{a})=n-1. \end{equation*}So, the result follows from applying the induction hypothesis to $\operatorname{ker}(g)$ .