Local cohomology with respect to an ideal $\mathfrak{a}$ is often studied over a Noetherian ring $R$. However, the proof of a lot of basic results does not rely on noetherianity of $R$, but rather on the following two properties:
(ITI) $\mathfrak{a}$-torsion submodules of injective modules are injective.
(ITR) $\mathfrak{a}$-torsion modules have injective resolutions whose components are $\mathfrak{a}$-torsion.
If $R$ is Noetherian, then it has ITI with respect to every $\mathfrak{a}$. If $R$ has ITI with respect to $\mathfrak{a}$, then it has ITR with respect to $\mathfrak{a}$.
Is anything known about the converses of these implications? Does anybody know a ring that does not satisfy ITR?
