Projective Modules

Quote:

Originally Posted by jcir2826

Prove that I_6 is simultaneously an injective and a projective module over itself.

As a $\mathbb{Z}_6$ -module, $\mathbb{Z}_6$ is free. Surely you know that all free modules are simultaneously projective. Thus, all we have to show is that $\mathbb{Z}_6$ is injective. To do this we can use Baer's criterion, which says that $\mathbb{Z}_6$ will be injective over $\mathbb{Z}_6$ if and only if we can extend every $\mathbb{Z}_6$ -homomorphism $\mathfrak{a}\to\mathbb{Z}_6$ can be extended to $\mathbb{Z}_6\to\mathbb{Z}_6$ for every $\mathfrak{a}$ a left ideal of $\mathbb{Z}_6$ . So...