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:



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> (ITI) $\mathfrak{a}$-torsion submodules of injective modules are injective.
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> (ITR) $\mathfrak{a}$-torsion modules have injective resolutions whose components are $\mathfrak{a}$-torsion.
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> 
> 


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?



injectivity of torsion submodules of injectives
@ Fred: I think the following may be help you to give an example for (ITR) question:


1. Choose a non-Noetherian local ring $(A, \frak{m})$ such that $\frak{m}= \frak{m}^n$ for all $n$.
2. Let $E(k)$ be the injective envelope of $k = R/\frak{m}$.
3. Claim: if $E(k)$ is $\frak m$-torsion, then $E(k) = k$. Indeed, let $x \in E(k)$. There is $n$ such that $\frak{m}^n x = 0$. Hence $\frak{m} x = 0$. But $k \subseteq Rx$, so $x \in k$. So $k$ is injective.


Dear Fred, I give here that a such ring.


Let $k$ be a file and $Q^+$ be the semigroup of non-negative ration.
Let $A = k[Q^+]$ be the semigroup ring, i.e


$A = \{ \sum\_{\alpha}u\_{\alpha}x^{\alpha}: u\_{\alpha} \in k \}$.


Let $\frak{n} = (x^{\alpha}: \alpha > 0)$ be the maximal ideal of $A$. Since for all $\alpha > 0$ we have $x^{\alpha} = x^{\alpha/2}x^{\alpha/2} \in \frak{n}^2$, so $\frak{n} = \frak{n}^2$.


Let $S = A\_{\frak{n}}$.
Let $I = (x^{\alpha}: \alpha > 1)$ be an ideal of $S$.
Let $(R, \frak{m})$ be the quotient ring $S/I$. We have $\frak{m} = \frak{m}^2$ by above.


**Claim:** $k$ is not injective.


**Proof of claim:** We consider the ideal $(x) \subseteq R$.
Notice that $(x) \cong k$ as $R$-modules. If $k$ is injective, then (x) is a direct summand of local ring $R$. It is a contradiction.


