Exercise 7, § 2

a) If u is a surjective endomorphism of an A-module E, u is not a right divisor of zero in $\mathrm{End}_A(E)$. Conversely, if for every submodule $F \neq E$ of E, there exists a linear form $x^* \in E^*$, zero on F and surjective, every element not a right divisor of zero in $\mathrm{End}_A(E)$ is a surjective endomorphism. b) If E is the $\mathbf{Z}$-module defined in exerc. 6 c), prove that every endomorphism $\neq 0$ of E is surjective, although there exists no linear form $\neq 0$ on E.

c) Prove that if E is a $\mathbf{Z}$-module free $\neq 0$, there exist non-surjective endomorphisms of E which are not right divisors of zero in $\mathrm{End}_{\mathbf{Z}}(E)$.