Can u help me prove the following statement
Statement about injective Modules
Proving the fact that if it is injective the sequence splits seems easy and i did it. Now the other way around ive tried using that fact that if the sequence splits then there exists inverse homomorphisms and i tried working with them but i didnt have any success. So any help is appreciated , thanks.
For the converse, if you have a diagram \begin{alignat}{3} 0\longrightarrow& M\longrightarrow N\\ &\downarrow \\ &\:I \end{alignat} consider the amalgamated sum $\;I\coprod\limits_M N$ and the canonical injection $$0\longrightarrow I\longrightarrow I\coprod\limits_M N,$$ which has a retraction by the hypothesis on $I$. Compose it with the canonical injection $N\longrightarrow I\coprod\limits_M N$.