## Every R-module is projective if and only if every R-module is injective

Let $R$ be a ring with 1. Show that every (unital, left) $R$-module is projective if and only if every (unital, left) $R$-module is injective.

Suppose first that every $R$-module is projective. Let $Q$ be an $R$-module.

Now let $0 \rightarrow Q \rightarrow M \rightarrow N \rightarrow 0$ be a short exact sequence. Since $N$ is projective, this sequence splits. So $Q$ is injective.

Conversely, if every $R$-module is injective and $P$ is a module, then every short exact sequence $0 \rightarrow L \rightarrow M \rightarrow P \rightarrow 0$ splits because $L$ is injective. So $P$ is projective.

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