Rank inequality for an injection of finite free modules
= Rank inequality for an injection of finite free modules
Over a nonzero <commutative ring>, an <injection> $R^m\hookrightarrow R^n$ implies $m\leq n$. 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.