Exercise 16, § 8

Let $\mathbf{E}, \mathbf{F}$ be two free $\mathbf{A}$-modules of respective dimensions $m$ and $n$. For a linear mapping $u : \mathbf{E} \to \mathbf{F}$ to be injective, it is necessary and sufficient that $m \leq n$ and that, if $X$ denotes the matrix of $u$ with respect to any two bases of $\mathbf{E}$ and $\mathbf{F}$, there exist no scalar $\mu \neq 0$ such that the products by $\mu$ of all the minors of $X$ of order $m$ are zero.