This is an exercise from Rotman , Introduction to homological algebra.
Given a pushout diagram in $R$-Mod
$$\begin{array}
AA & \stackrel{g}{\longrightarrow} & C \\
\downarrow{f} & & \downarrow{\beta} \\
B & \stackrel{\alpha}{\longrightarrow} & D
\end{array}
$$
prove that $g$ injective implies $\alpha$ injective, and that $g$ surjective implies $\alpha$ surjective.
I have problems with the injective part, how to solve it ?
Best Answer
So I applied the construction of pushout in $R$-Mod: $$D \cong (B \oplus C)/S$$ where $S$ is the submodule generated by $$\lbrace (f(a), -g(a) ) | \ a \in A \rbrace$$
Thus if $\alpha(b) = 0 $ we have that $$\alpha(b) = (b,0) \in S \Rightarrow \exists a \in A \ \text{s.t} \ \ (b,0) = (f(a) , -g(a))$$
But this implies $a= 0$ due to injectivity of $g$ and then $b= 0$.