I would like to prove the following:
Let $R$ be a commutative Noetherian ring with identity, $I$ an injective module, and $N=\sqrt{(0)}$ (the nilradical). Then $I/NI$ is an injective $R/N$ module.
If I can show that the injective dimension of $NI$ over $R$ is less than or equal to 1 I think I know how to prove it via the short exact sequence
$0\to NI\to I\to I/NI$.
Thanks for the help.