Exercise 8, § 2

(a) Let E be a free A-module, F, G two A-modules and $u : E \to G$, $v : F \to G$ two linear mappings. Show that if $u(E) \subset v(F)$, there exists a linear mapping $w : E \to F$ such that $u = v \circ w$.

(b) Give an example of two non-zero endomorphisms $u, v$ of the $\mathbf{Z}$-module E defined in Exercise 6(c) such that there exists no endomorphism $w$ of E for which $u = v \circ w$ (although $u(E) = v(E) = E$).

(c) Give an example of two endomorphisms $u, v$ of the $\mathbf{Z}$-module $\mathbf{Z}$ such that $-u^{-1}(0) = -v^{-1}(0) = \{0\}$, but such that there exists no endomorphism $w$ of $\mathbf{Z}$ for which $u = w \circ v$ (cf. § 7, Exercise 14).