I'm doing the following exercices about R-modules and superfluous epimorphism
Consider the following conmutative diagram in cathegory of $\require{AMScd}R$-modules, asume that both rows are exact: Exact Rows like this
\begin{CD} 0 @>>> A @>f>> B @>g>> C @>>> 0\\ & @V \alpha VV @V \beta VV @V \gamma VV\\ 0 @>>> A' @>f'>> B' @>g'>> C' @>>> 0 \end{CD}
- If $\alpha$ is surjective and Ker g is superfluous then Ker g' is superfluous.
- If $\gamma$ is injective and Ker g' is superfluous then Ker g is superfluous.
For 1. $\text{Ker} g = \text{Im} f$ is superfluous so $\beta(f(A))$ is superfluous in $B'$, but $\beta(f(A))= f'(\alpha(A))$ and $\alpha$ is sujerctive so $\alpha(A) = A'$, so $\beta(f(A))= f'(A') = \text{Im} f' = \text{Ker} g'$ so $\text{Ker} g'$ is superfluous.
For 2. i want to show that $\text{Ker} g$ is superfluous, i found $\text{Ker} \beta \subseteq \text{Ker} g$ and i prove this general result: $\text{Ker} g$ is superfluous if and only if for all $h: D \rightarrow B$ such that $g \circ h$ is surjective then $h$ is surjective but i dont know how to prove 2.
Thanks.
2 is clearly false. For example, take $C=C'$ and $\gamma=\text{id}$. Fix a map $g'\colon B'\to C'$ such that $\text{Ker}(g')$ is superfluous. Choose $B$ to be a direct sum of $B'$ and a non-zero module, say $F$, and $\beta$ to be the projection of $B$ onto $B'$. $g$ is simply the composision $g'\beta$. Since $\text{Ker}(g)$ contains $F$, it can't be superfluous.