Exercise 5, § 1

Let E, $F_1, F_2$ be three A-modules and $f_1 : E \to F_1, f_1 : E \to F_2$ two linear mappings. The amalgamated sum of $F_1$ and $F_2$ relative to $f_1$ and $f_2$ is the quotient module of $F_1 \times F_2$ by the submodule the image of E under the mapping $z \mapsto (f_1(z), -f_2(z))$; it is denoted by $F_1 \oplus_E F_2$.

(a) For every ordered pair of A-linear mappings $u_1 : F_1 \to G, u_2 : F_2 \to G$ such that $u_1 \circ f_1 = u_2 \circ f_2$, show that there exists one and only one A-linear mapping $u : F_1 \oplus_E F_2 \to G$ such that $u \circ j_1 = u_1, u \circ j_2 = u_2$, where $j_1$ and $j_2$ denote the respective compositions of the canonical mapping

with the canonical injections $F_1 \to F_1 \times F_2, F_2 \to F_1 \times F_2$.

(b) Let $E', F'_1, F'_2$ be three A-modules, $f'_1 : E' \to F'_1, f'_2 : E' \to F'_2$ two A-linear mappings and $F'_1 \oplus_{E'} F'_2$ the amalgamated sum relative to these mappings. For every system of A-linear mappings $v_1 : F'_1 \to F_1, v_2 : F'_2 \to F_2, w : E' \to E$ such that $v_1 \circ f'_1 = f_1 \circ w, v_2 \circ f'_2 = f_2 \circ w$, let $v$ be the A-linear mapping of $F'_1 \oplus_{E'} F'_2$ into $F_1 \oplus_E F_2$ corresponding to the two linear mappings $j_1 \circ v_1$ and $j_2 \circ v_2$. Show that if $v_1$ and $v_2$ are surjective, so is $v$; give an example where $v_1$ and $v_2$ are injective but $v$ is not injective. Show that if $w$ is surjective and $v_1$ and $v_2$ injective, then $v$ is injective.