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In the paper "Almost diagonal matrices over Dedekind Domains" by L. Levy a specific decomposition of modules over Dedekind rings $D$ is used:

Let $M$ be submodule of $D^n$, then there exists a simultaneous decomposition

$D^n \cong D y_1 \oplus \dots \oplus D y_{r-1} \oplus H^{-1} \oplus H \oplus D^{n-r-1}$

$M \cong L_1 y_1 \oplus \dots \oplus L_{r-1} y_{r-1} \oplus L_r H^{-1}$

where $L_i$are integral ideals and $H$ a fractional ideal, $H^{-1}$ being its inverse.

This decomposition is not proved, Levy just mentions that it is scattered through the works of Krull. I want to ask, if someone knows a reliable (preferably modern) reference for this decomposition.

Furthermore, is it possible to derive it from the "ordinary" decomposition of f.g. modules $N$ over Dedekind domains (which can for instance be found in Narkiewicz book)?: $N \cong \bigoplus_{k=1}^s R/a_i \oplus H \oplus R^k$ where the $a_i$ are ideals of $R$.

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