Finitely generated torsion-free module over a principal ideal domain

Every finitely generated torsion-free module over a principal ideal domain is free. In the structure theorem, torsion-freeness removes every cyclic torsion summand and leaves only the free summand.

An integral matrix satisfying $A^2+A+I=0$ makes ${\mathbb{Z}}^n$ a module over the Eisenstein integers by letting the primitive cube root act as $A$. This module is finitely generated and torsion-free, hence free. Since the Eisenstein integers have integer rank two, such matrices exist only for $n=2m$; for that dimension every one is integrally conjugate to