Let be a ring with 1. Show that every (unital, left) -module is projective if and only if every (unital, left) -module is injective.
Suppose first that every -module is projective. Let be an -module.
Now let be a short exact sequence. Since is projective, this sequence splits. So is injective.
Conversely, if every -module is injective and is a module, then every short exact sequence splits because is injective. So is projective.