Show that the following are equivalent for a ring:
(1) any $R$-module is projective.
(2) any $R$-module is injective
This is problem in Dummit Foote book, problem number 6, page 403. I was trying by using equivalent definition of projective and injective module but nothing find effective.Any help/hint in this regards would be highly appreciated. Thanks in advance!
A module $M$ is projective iff every short exact sequence $$0\to A\to B\to M\to0$$ splits. So all modules are projective iff every short exact sequence splits. So what about injectivity instead of projectivity ?