Question
Let $E$ be a left $R$-module. Prove that $E$ is injective if and only if $\operatorname{Ext}_{R}^{1}(A, E)=\{0\}$ for every left $R$-module $A$.
Step 1
A left $R$-module $E$ is injective if for every left $R$-module $A$ and every injective homomorphism $f: A \to B$, and every homomorphism $g: A \to E$, there exists a homomorphism $h: B \to E$ such that $h \circ f = g$. Show more…
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