Prove that I_6 is simultaneously an injective and a projective module over itself.
As a -module, is free. Surely you know that all free modules are simultaneously projective. Thus, all we have to show is that is injective. To do this we can use Baer's criterion, which says that will be injective over if and only if we can extend every -homomorphism can be extended to for every a left ideal of . So...