Exercise 1, § 1

Let E, F be two A-modules and $u : E \to F$ a linear mapping. Let M be a submodule of E, N a submodule of F, $i : M \to E$ the canonical injection and $q : F \to F/N$ the canonical surjection. Show that $u(M) = \operatorname{Im}(u \circ i)$ and $u^{-1}(N) = \operatorname{Ker}(q \circ u)$.