Basis of a subset of finitely generated torsion free module

Based on the comments of rschwieb’s answer in this question asked recently: Can we contruct a basis in a finitely generated module.

If $M=\langle e_1,\ldots,e_n\rangle$ is a finitely generated torsion free $R$-module. I’m trying to construct a free submodule $F$, i.e, isomorphic to $R^s$ for some $s$, finding a subset $S=\{e_1,\ldots,e_s\}$ such that $S$ is a maximal independent subset of $M$, then $S$ generates this free submodule $F$ of $M$ with basis $S$.

I’m asking that because I didn’t understand why Peter Clark in his commutative algebra pdf wrote this:

Thanks in advance

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