Corollary 2

— Every torsion free finitely generated module over a principal ideal domain is free of finite rank. This follows immediately from Cor. 1.

The condition that the module be finitely generated is essential. For example the additive group of the field of fractions K of A is torsion free as an A-module; however it is not a free module if $A \neq K$, for on the one hand any two elements of K have a common divisor, and on the other hand K is not a cyclic A-module, for otherwise $K = ab^{-1}A$ ($a, b \in A$), whence $b^{-2} = acb^{-1}$ ($c \in A$), so $b^{-1} = ac \in A$, and $K = A$.