# Hermite normal form and saturation

Recall that if $M$ is a submodule of $\mathbb{Z}^n$, then the saturation of $M$ (in $\mathbb{Q}$) is defined to be $\mathbb{Z}^n \cap (\mathbb{Q}\otimes_{\mathbb{Z}} M)$. According to an article of Pernet and Stein [1, Section 8], there is a well known connection between the saturation of $M$ and the Hermite normal form of the basis matrix of $M$. Specifically, let $A$ be the basis matrix of $M$, then a basis matrix for the saturation of $M$ is given by $H^{-1}A$ where $H$ is the Hermite normal form of the transpose of $A$.

Unfortunately there's no citation given for this result in [1], and, try as I might, I have had no luck finding any other reference to this result anywhere. Does anyone know of one?

[1] Pernet, Clément; Stein, William, "Fast computation of Hermite normal forms of random integer matrices". J. Number Theory 130 (2010), no. 7, 1675–1683. Also available here: http://modular.math.washington.edu/papers/hnf/hnf.pdf

-