# Questions tagged [injective-modules]

For questions about injective modules over a ring and injective objects in related categories.

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### Freeness of injective hull of finite module over Gorenstein Artinian ring

This is Lemma 4.1 in Brochard, Khare, Iyengar: Wiles defect and criteria for freeness. I have managed to understand the proof, except for the following part: Why is $F$, the injective hull of a finite ...
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### Pontrjagin dual of modules [closed]

I am not sure whether this question is appropriate to appear here. If not, I apologize for that. Given an $R$-module $M$, we define its Pontrjagin dual as $M^{\ast}=Hom_{\mathbb{Z}}(M, \mathbb{Q/Z})$. ...
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### Prove that $f(M)=f^2(M)$ implies $f(M)$ is a direct summand of $M$ whenever $\text{End}_R(M)$ is a reduced ring

Let $M$ be a right $R$-module with the property that every homomorphism $\gamma:Sf\to M, f\in S=\text{End}_R(M)$, extends to $S\to M$. If $S$ has the property $f^2=0$ implies $f=0$ for every $f\in S$...
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### Looking for example of quotient of group algebra by ideal of group ring which fails to be injective

I am looking for an example of a group ring $\mathbb{Z}[G]$ of a finite group $G$ along with a lattice $I$ (in the case at hand the word 'lattice' means: a $\mathbb{Z}[G]$-submodule which is ...
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### Why does there exist a non-split sequence with the condition that $\mathrm{pd} M=\infty$?

Why does there exist a non-split sequence with the condition that $\mathrm{pd} M = \infty$? Remarks. I am reading Andrzej Skowroński, Sverre O.Smalø, Dan Zacharia: On the Finiteness of the Global ...
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### Direct sum of K-injectives over a noetherian ring

Let $A$ be a noetherian ring, and let $\{I_i \mid i \in J\}$ be a collection of K-injective complexes over $A$. Is the direct sum $$\bigoplus_{i \in J} I_i$$ also a K-injective complex over $A$? ...
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### Ring such that any submodule of an injective module is flat?

Does anyone know examples of rings $R$ with the property that any submodule of an injective (right) $R$-module is flat? If I'm not missing something, this class of rings includes the (Von Neumann) ...
Let $R$ be a noetherian local ring with maximal ideal $\mathcal m$ and denote by $E$ the injective hull of the residue field $k$. Then, as an $R-$module, what is the support of $E$?
Do you know of any self-injective basic algebra $A$ over a field $k$ such that its enveloping algebra $A^{\mathrm{op}}\otimes_k A$ is not self-injective? The algebra $A$ cannot be finite-dimensional, ...