Rank of a free module ($\mathrm{rk}M$)

For a nonzero commutative scalar ring, every two bases of a finite-rank free module have the same cardinality. This cardinality is the rank.

Over a nonzero commutative ring, an injection $R^m\hookrightarrow R^n$ implies $m\le n$. If $m>n$, appending zero coordinates gives an injective endomorphism of $R^m$ with zero determinant; the Cayley--Hamilton theorem and cancellation of that injection eventually force the identity to vanish.