Exercise 8, § 2

a) Let E be a free a-module, F, G two A-modules, $u : E \to G$, $v : F \to G$ two linear mappings. Now prove that if one has $u(E) \subset v(F)$, there exists a linear mapping $w : E \to F$ such that $u = v \circ w$. b) Give an example of two nonzero endomorphisms $u, v$ of the $\mathbf{Z}$-module E defined in Exercise 6 c) of II, p. 183, such that there exists no endomorphism $w$ of E for which $u = v \circ w$ (though $u(E) = v(E) = E$). c) Give an example of two endomorphisms $u, v$ of the $\mathbf{Z}$-module $\mathbf{Z}$ such that $\overline{u}^1(0) = \overline{v}^1(0) = \{0\}$, but such that there exists no endomorphism $w$ of $\mathbf{Z}$ for which $u = w \circ v$ (cf. II, p. 195, Exercise 14).