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An $A$-module $N$ is injective if and only if $\text{Ext}_{A}^1(M,N)=0$ for any $A$-module $M$. I asked a similar question and got a hint but I am still stuck, this is how far I have got:

If $N$ is injective we simply consider the injective resolution $$0 \to N \to N \to 0 \to 0 \to \dots $$ from here it is easily seen that $\text{Ext}_{A}^1(M,N)=0$ for any module $M$. IS this correct? I am stuck on the other implication.

Suppose that $0 \to M_1 \to M_2$ is an injection, how can I then see that $$\text{Hom}(M_2,N) \to \text{Hom}(M_1,N) \to 0$$ is a surjection using the fact that $\text{Ext}_{A}^1(M,N) = 0$ for any $M$?

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if $0\rightarrow M_1\rightarrow M_2\rightarrow C\rightarrow 0$,then we can get $0\rightarrow Hom(C,N) \rightarrow Hom(M_2,N)\rightarrow Hom(M_1,N)\rightarrow Ext^1(C,N)$ is exact sequence.If N is injective,then $Ext^1(C,N)=0$. The basic definiton of injective module is also this question you ask is satisfied.

Jian
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