We have a famous theorem for a vector space which states that "every set of generators of a vector space $V$ over a field $F$ contains a basis of $V$". Is this statement true for a free module (probably over a PID) instead of the vector space $V$?


closed as off-topic by user26857, C. Falcon, Namaste, user91500, Ali Caglayan Dec 20 '16 at 6:18


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  • $\begingroup$ Is this statement true...? $\endgroup$ – user317953 Dec 19 '16 at 19:24
  • $\begingroup$ What do you think, and why? $\endgroup$ – Namaste Dec 20 '16 at 0:46

No. Consider the set $\{2,3\}$, which generates $\mathbb{Z}$ as a $\mathbb{Z}$-module; it does not contain a basis of $\mathbb{Z}$.

