Czechoslovak Mathematical Journal, first online, pp. 1-12
A characterisation of a class of $\mathbb{Z}$-saturated finitely generated submodules of free $\mathbb{Z}[X]$-modules
Sidi Mohamed EL MOUSTAPHA, Ihsen YENGUI
Received February 10, 2026. Published online September 18, 2026.
Abstract: It is known that to every $\mathbb{Z}$-saturated submodule of $\mathbb{Z}^m$ one can associate a unique nonnegative integer (an invariant) which is equal to its dimension as $\mathbb{Q}$-vector space and also as $\mathbb{F}_p$-vector space for any prime number $p$. The main goal is to generalize this result to any $\mathbb{Z}$-saturated submodule of $\mathbb{Z}[X]^m$ generated by $\mathbb{Z}[X]$-linearly independent vectors (for example, a syzygy module over $\mathbb{Z}[X]$ whose iterated syzygy module is zero). In terms of matrices, we prove that the image of any matrix $A \in{\cal M}_{m \times r}(\mathbb{Z}[X])$ whose columns are $\mathbb{Z}[X]$-linearly independent is $\mathbb{Z}$-saturated if and only if the integer $r$ which is equal to its rank over $\mathbb{Q}[X]$, i.e. when it is seen in ${\cal M}_{m \times r}(\mathbb{Q}[X])$, is also equal to its rank over univariate polynomial rings over the prime fields $\mathbb{F}_p$, i.e., when $A$ is seen in ${\cal M}_{m \times r}(\mathbb{F}_p[X])$. To do so, we begin by addressing the local case (the base ring being a valuation domain). Our proofs are both simple and constructive.
Affiliations: Sidi Mohamed El Moustapha, Ihsen Yengui (corresponding author), Departement of Mathematics, Faculty of Sciences, University of Sfax, Road Soukra 1171, 3000 Sfax, Tunis, email: hattabsidi697@gmail.com, ihsen.yengui@fss.rnu.tn
