Lemma 9.4.1. If $k$ is a field, then every $k$-module is free.

Proof. Indeed, by linear algebra we know that a $k$-module (i.e. vector space) $V$ has a basis $\mathcal{B} \subset V$, which defines an isomorphism from the free vector space on $\mathcal{B}$ to $V$. $\square$

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