Exercise 6, § 2

(a) If E is an A-module, every injective endomorphism u of E is not a left divisor of zero in the ring $\operatorname{End}_A(E)$. Conversely, if for every sub-A-module $F \neq \{0\}$ of E there exists an endomorphism $v \neq 0$ of E such that $v(E) \subset F$, an element which is not a left divisor of zero in $\operatorname{End}_A(E)$ is an injective endomorphism. The above condition is satisfied if there exists a linear form $x' \in E^*$ and an $x \in E$ such that $\langle x, x' \rangle$ is invertible in A and in particular if E is free.

(b) Every endomorphism $\neq 0$ of the $\mathbf{Z}$-module $\mathbf{Q}$ is bijective, although there exists no linear form $\neq 0$ on $\mathbf{Q}$.

(c) Let $U_p$ be the $\mathbf{Z}$-module defined in § 1, Exercise 12(d). Show that the endomorphism $x \mapsto px$ of $U_p$ is not injective and is not a left divisor of zero in $\operatorname{End}_{\mathbf{Z}}(U_p)$.