Exercise 4, § 1

Let E, $F_1, F_2$ be three A-modules and $f_1 : F_1 \to E, f_2 : F_2 \to E$ two A-linear mappings. The fibre product of $F_1$ and $F_2$ relative to $f_1$ and $f_2$ is the submodule of the product $F_1 \times F_2$ consisting of the ordered pairs $(x_1, x_2)$ such that $f_1(x_1) = f_2(x_2)$; it is denoted by $F_1 \times_E F_2$.

(a) For every ordered pair of A-linear mappings $u_1 : G \to F_1, u_2 : G \to F_2$ such that $f_1 \circ u_1 = f_2 \circ u_2$, there exists one and only one A-linear mapping $u : G \to F_1 \times_E F_2$ such that $p_1 \circ u = u_1, p_2 \circ u = u_2$, where $p_1$ and $p_2$ denote the restrictions to $F_1 \times_E F_2$ of the projections pr_1 and pr_2.

(b) Let $E', F'_1, F'_2$ be three A-modules, $f'_1 : F'_1 \to E', f'_2 : F'_2 \to E'$ A-linear mappings and $F'_1 \times_{E'} F'_2$ the fibre product 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 $f'_1 \circ v_1 = w \circ f_1, f'_2 \circ v_2 = w \circ f_2$, let $v$ be the A-linear mapping of $F_1 \times_E F_2$ into $F'_1 \times_{E'} F'_2$ corresponding to the two linear mappings $v_1 \circ p_1$ and $v_2 \circ p_2$. Show that if $v_1$ and $v_2$ are injective, so is $v$; give an example where $v_1$ and $v_2$ are surjective but $v$ is not surjective (take $E' = \{0\}$). If $w$ is injective and $v_1$ and $v_2$ surjective, show that $v$ is surjective.