How to prove this equivalent condition to that B is an injective R-module?

Question

B is an injective R-module iff $ Ext^{1}_{R}(A,B) $ vanishes for all A. I know how to prove this statement from left to right, but don't know the opposite direction. Please help.

Answer 1

$B$ is injective if and only if the functor $Hom_R(-,B)$ is exact. Take a short exact sequence:

$$ 0 \rightarrow A' \rightarrow A \rightarrow A'' \rightarrow 0 $$

And apply the functor $Hom_R(-,B)$ to get an exact sequence:

$$ 0 \rightarrow Hom_R(A'',B) \rightarrow Hom_R(A,B) \rightarrow Hom_R(A',B) \rightarrow Ext_R^1(A'',B) \rightarrow Ext_R^1 (A,B) \cdots $$

By hypothesis $Ext_R^1(A'',B)=0$, then you have an exact sequence:

$$ 0 \rightarrow Hom_R(A'',B) \rightarrow Hom_R(A,B) \rightarrow Hom_R(A',B) \rightarrow 0 $$

Then $B$ is injective.

If you are using the diagram definition of injective, try to prove what I said on the first line.