$\Gamma_I(E)$ is an injective $R$-module? $H^i_I(E)=0;\forall i\gt 0$

Question

1.Let $R$ be a commutative ring, $M$ an $R$-module, $I$ an ideal in $R$, and $E$ an injective $R$-module. Can one claim that $H^i_I(E)=0;\forall i\gt 0$?

2.In the case of noetherian rings we know that $\Gamma_I(E)$ is an injective $R$-module. Is it also true for non-noetherian rings that "$\Gamma_I(E)$ is an injective $R$-module"? (or there is a counter example?)

Thank you.

Answer 1

Concerning Question 2: This does not hold if you just omit the noetherian hypothesis. The situation in general is rather complicated. You can take a look at this MO question and some of my other questions or answers there, while waiting for mine and Quy's paper to be published...

Answer 2

$H_I^i$ is a right-derived functor of a left-exact functor. By construction any injective module is acyclic with respect to that functor. So $H_I^i(E)=0$ holds trivially for $E$ injective.