Let $R$ be a local ring. We know, for example, that $V$ is a projective $R$ module if and only if $V$ is free. Also, since $R$ is not semisimple, there must be at least one module which is not projective and one which is not injective.
Is there a way to classify injective modules over a local ring? The best candidate would be the stament "$V$ is injective if and only if it is divisible." However, for example, $\Bbb Z/(3)$ is divisible as a $\Bbb Z/(9)$ module, but not injective since $\Bbb Z/(9)\not\cong \Bbb Z/(3)\oplus \Bbb Z/(3)$.
My only reason for believing there is such a statement is because the notions are categorically dual. Still, that doesn't imply that there must be some analog, but I thought I'd ask.