let $i: B'\rightarrow B$ and $p: B\rightarrow B''$ be $R-maps$, where $R$ is a ring. If for every $left\ R-module$ M,
$0\rightarrow Hom_R(B'',M)\rightarrow Hom_R(B,M)\rightarrow Hom_R(B',M)\rightarrow0$
is an exact sequence of abelian groups, induced by $i$,$p$.
show that
$0\rightarrow B'\rightarrow B\rightarrow B''\rightarrow 0$
is a split short exact sequence.
This question come from the Rotman's An introduction to homological algebra.
On his book, this sequnce is exact $B'\rightarrow B\rightarrow B''\rightarrow 0$.
To showing that $i$ is injective. let $f:B'\rightarrow B'/ker i$. How to get that $kerf$ is $0$.