Suppose $R$ is a Dedekind domain and $A$ an $m \times n$ matrix of rank $r$ over $R$. Then $A$ induces an $R$-module homomorphism $\varphi:R^m \to R^n$ via $x \mapsto xA$ giving rise to an exact sequence $$R^m \xrightarrow{\alpha} R^n \to S:=R^n/\operatorname{im}\varphi\to 0,$$ i.e. $S$ is a finitely presented $R$-module.
A finitely generated module $M$ over $R$ may be decomposed into a torsion part and a torsion-free part: $M = \mathcal T(M) \oplus \mathcal{TF}(M)$.
Is it true that the minimal number of $R$-generators for $\mathcal T(S)$ is equal to $r$ and the minimal number of generators for $\mathcal{TF}(S)$ equals $n-r$?