Modules: monos are stable under pushouts

Question

In $R$-Mod, monos are stable under pushouts: suppose in $R$-Mod that $f_1:M \rightarrowtail M_1$ is a mono and $f_2:M\to M_2$ so that they form a span. Complete this to a pushout $\hat{f}_2:M_1\to N$ and $\hat{f}_1:M_2\to N$. I want to show that $\hat{f}_1:M_2\rightarrowtail N$ is in fact a mono. Can a detailed calculation of this fact be given ?

Answer 1

$N$ is constructed as the quotient of $M_1\oplus M_2$ by the submodule $I=\{(f_1(m),f_2(m)):m\in M\}$. If $\hat f_1(m_2)=0$, then $(0,m_2)$ must be in $I$. This implies that $m_2=f_2(m)$ with $f_1(m)=0$. But then since $f_1$ is mono, $m$ and thus $m_2$ must be $0$.