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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