Let $R$ be a local ring with maximal ideal $J$. Assume that ${\rm End}_{R}({\rm E}(R/J))$ is a division ring (${\rm E}(R/J)$ means the injective envelope of $R/J$). Does $R/J$ is injective?
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$(R/J)_R$ will be injective only in trivial cases.
For a local ring, the only simple module is $R/J(R)$. If that is an injective right module, then it is a right V-ring, but V-rings are all semiprimitive, so it would mean $R$ is already a division ring.
So if you have a particular example in hand where $J\neq \{0\}$, you can rest assured $R/J$ is not an injective $R$ module.
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$\begingroup$ Thank you very much for your comment. This is right. In fact I am searching for an example. However in commutative case, I proved that if $R$ is a local ring and ${\rm End}_{R}(E(R/J))$ is division ring then $R$ must be a field. But in general case, I do not know. $\endgroup$ Commented Oct 24, 2023 at 15:13