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Let $M$ be a finitely generated $R$-module with rank $n$, and let $N = \left\{ y_{1},\ldots,y_{m}\right\}$ be an $R$-linearly independent subset of $M$, where $m \leq n$.

My question is simple: Is it true that $N$ is contained in a maximal linear independent subset of $M$? I guess the answer is yes with the help of Zorn's lemma. If this is the case, is there a reference for this?

Any help/hint is highly appreciated!

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  • $\begingroup$ Please clarify what you mean by rank. $\endgroup$ – user26857 Aug 13 '18 at 5:54
  • $\begingroup$ In any case, you may assume that N is not maximal and add new elements one by one. This way you get more than n independent elements which is not possible. $\endgroup$ – user26857 Aug 13 '18 at 5:58

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