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# Questions tagged [injective-module]

A module $I$ over a ring $R$ is injective if $\hom_{R}({-},I)$ is exact. The notion of injective modules is dual to the notion of a projective module. In homological algebra injective modules are used for computing right derived functors.

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### Is there a Baer's criterion for testing injectivity of sheaves of $\mathcal{O}_X$-modules?

In the important paper by Spaltenstein on resolving unbounded complexes, they turn their hand to sheaves. Let us fix a ringed space $(X;\mathcal{O}_X)$. In the proof of Lemma $4.3$ it is implicitly ...
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### ${\mathrm{Ext}}_{k[\epsilon]}^1(k, k)$ and ${\mathrm{Ext}}_{k[X]}^1(k, k)$.

For a field $k$, I am calculating ${\mathrm{Ext}}_{k[\epsilon]}^1(k, k)$ and ${\mathrm{Ext}}_{k[X]}^1(k, k)$, where $\epsilon^2 = 0$. However, there seems no complete explanation as far as I checked. ...
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### $\operatorname{Hom}_k(R,k)$ is injective indecomposable $R$-module for $R$ local $k$-algebra of finite $k$-dimension

I am working on Exercise 3.1.22. from Bruns and Herzog's Cohen-Macaulay rings. The exercise in question is the following (the references I will use are of course from within the book): It appears ...
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### Equivalent conditions of $\textrm{Ext}^2(A,B) = 0$ for all $A$ and $B$

I am studying homological algebra and I am having difficulty proving the equivalences of the following: (i) If $0 \to A \xrightarrow{f} B$ is exact and $B$ is projective, then $A$ is projective (ii) ...
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### Proposition 13, Section 4.3 of Hungerford’s Algebra

Proposition 3.12. Every unitary module $A$ over a ring $R$ with identity may be embedded in an injective $R$-module. Proposition 3.7. A direct product of $R$-modules $\prod_{i\in I} J_i$ is injective ...
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### Is an injective module an injective object in every full subcategory that contains it?

Let $\mathcal{C}$ be a full subcategory of $\text{Mod}_A$, the category of modules over a commutative unital ring $A$ (I am mostly interested in the graded setting, but I doubt there is much ...
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### Computing the injective hull, the quasi-injective hull, the quasi-continuous hull

Let $M_R$ be any module. Recall that $M_R$ is called quasi-injective if every $R$-homomorphism $N\to M$ from a submodule $N$ of $M$ extends to $M$, $M_R$ is called quasi-continuous if it's both CS (i....
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### What is the injective hull of the ring $R=\mathbb{R}[x]/(x^2)$?

Let $R$ be a ring with unity and $M$ a right $R$-module. Recall that the injective hull $E(M_R)$ of $M_R$ is the maximal essential minimal injective extension of $M_R$. Let $R=\mathbb{R}[x]/(x^2)$. ...
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### Finiteness of the Injective Hull

Please help me. I have read Finiteness of the Injective Hull by Alex Rosenberg and Daniel Zelinsky. An $R$-module $N$ has finite length if $N$ has a finite composition series. Alex Rosenberg and ...
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### Cardinal of the injective hull

Let $A$ be an abelian group that is countable. Is it true that the injective hull $E(A)$ of $A$ (as $\mathbb{Z}$-module) is also countable? For example, $E(\mathbb{Z})=\mathbb{Q}$ and for every prime ...
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### Indecomposable injective modules

Let $R$ be a ring with $1$. It's well known that if $M$ an indecomposable injective right $R$-module, then $M\cong E(R/\mathfrak{p})$ for some prime ideal $p\subset R$ where $E(R/\mathfrak{p})$ is the ...
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### Injective modules in short exact sequences

Let me first state the question itself. Let $R$ be a ring with $1$. Let $0\to M\to Q\to L\to 0$ and $0\to M\to Q'\to L'\to 0$ be short exact sequences of $R$-modules where $Q$ and $Q'$ are both ...
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### Injectivity of $\mathbb{Q}_{p}$ and $\mathbb{Z}(p^{\infty})$

Please help me. In the category of $R$-modules, we know that the Prüfer group $\mathbb{Z}(p^{\infty})$ is an injective envelope of $\mathbb{Z}/p\mathbb{Z}$ the ring integers modulo $p$ where $p$ is a ...
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### Two equivalent definitions of injective modules

According to my lecture notes, these two definitions of an injective module are equivalent: (Let $R$ be a ring, $Q$ an $R$-module.) For every injective $R$-module homomorphism $u:M’ \rightarrow M$ ...
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