Exercise 18, § 8

Let $\mathbf{E}$ and $\mathbf{F}$ be two free $\mathbf{A}$-modules of respective dimensions $m$ and $n$, $u : \mathbf{E} \to \mathbf{F}$ a linear mapping and $X$ the matrix of $u$ with respect to any two bases of $\mathbf{E}$ and $\mathbf{F}$. (a) If $m \geq n$ and there exists a minor of $X$ of order $n$ which is invertible, $u$ is surjective. (b) Conversely, if $u$ is surjective, show that $m \geq n$ and that there exists a minor of $X$ of order $n$ which is non-zero. If further, in the ring $A$, the ideal generated by the non-invertible elements is distinct from $A$, there exists a minor of $X$ of order $n$ which is invertible (consider the exterior power $\bigwedge^n u$).

(c) Let $B$ be a commutative ring and $A$ the product ring $B \times B$. Give an example of a surjective linear mapping of the $A$-module $A^2$ onto $A$, all the elements of whose matrix (with respect to the canonical bases of $A^2$ and $A$) are divisors of zero.