Why is a finitely generated Torsion-free module free?

Question

Lang-Algebra p.148

Let $R$ be a PID. Let $M$ be a finitely generated Torsion-free $R$-module. Let $\{v_1,...,v_n\}$ be a maximal set of elements of $M$ among a given finite set of generators $\{y_1,...,y_m\}$ such that $\{v_1,...,v_n\}$ is linearly independent.

I don't get the bolded sentence. What does it exactly mean? Since $M$ is finitely generated, there exists a finite subset $S\subset M$ such that $M=\langle S \rangle$. Does $\{v_1,...,v_n\}$ denote the maximal linearly independent subset of $S$?

Answer 1

It denotes a maximum linearly independent subset of $S$. In other words, it is a subset of $S$ such that, adding any element of $S$, one attains a linearly dependent set.

Such a subset exists since any singleton has this property (the module being torsion-free). One does not know (yet) that such any two such subsets have the same cardinality, say, but that will follow from the proof.