Exercise 7, § 2

(a) If u is a surjective endomorphism of an A-module E, u is not a right divisor of zero in $\operatorname{End}_A(E)$. Conversely, if for every submodule $F \neq E$ of E there exists a linear form $x^* \in E^*$ which is zero on F and surjective, every element of $\operatorname{End}_A(E)$ which is not a right divisor of zero is a surjective endomorphism.

(b) If E is the $\mathbf{Z}$-module defined in Exercise 6(c), show that every endomorphism $\neq 0$ of E is surjective, although there exists no linear form $\neq 0$ on E.

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