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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.

Someone
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1 Answers1

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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 $N\longrightarrow I\coprod\limits_M N$.

Bernard
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  • Im not familiar with the concept of an amalgamated sum and i didnt define it in my algebra course so i dont know :( , but thanks for the answer anyway. – Someone Dec 27 '18 at 17:17
  • Did you see fiber products? It's the dual notion. – Bernard Dec 27 '18 at 17:38
  • No sorry, im afraid i didnt go that far :/ – Someone Dec 27 '18 at 17:39
  • The amalgamated sum isnt't very complex: with the present notations, you consider the direct sum $I\oplus N$, and quotient out by the submodule consisting of $\bigl(f(m), -u(m)\bigr)$ for all $m\in M$, where $f:M\to I$ and $u:M\to N$. – Bernard Dec 27 '18 at 17:53
  • Alright it doesnt seem very complicated , ill take a look into it , thx – Someone Dec 27 '18 at 17:54