The rank of a finite free module is well defined.

Lemma 10.15.8. Let $R$ be a nonzero ring. Let $n, m \geq 0$ be integers. If $R^{\oplus n}$ is isomorphic to $R^{\oplus m}$ as $R$-modules, then $n = m$.

Proof. Immediate from Lemma 10.15.7. $\square$

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