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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Matlis dual of injective module over complete local ring [closed]

Let $M$ be an injective module over a Noetherian complete local ring $(R,\mathfrak m,k)$. Let $E(k)$ denote the injective hull of $k$. Then, is it true that $\text{Hom}_R(M, E(k))$ is a free $R$-...
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Proving second injective change of rings theorem

I am trying to prove the second injective change of rings theorem:$\DeclareMathOperator{\id}{id}$ Let $R$ be a ring, $A$ be an $R$-module, $x\in R$ a central non-unit non-zerodivisor such that $xa\...
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If $f:M\to M,m\mapsto mr$ is injective, then show that $\text{Hom}_R(M,E)\stackrel{r}{\longrightarrow}\text{Hom}_R(M,E)$ is surjective

Let $R$ be a commutative ring, $M$ an $R$-module and $r\in R$. If $f:M\to M$ defined by $f(m)=mr$ is an injective $R$-module endomorphism, then show that the mapping $\text{Hom}_R(M,E)\stackrel{r}{\...
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Prove that $\Bbb Q$ is an injective object in $\Bbb Z\text{-mod}$ using the following definition.

How can I show that $\Bbb Q$ is an injective object in $\Bbb Z\text{-mod}$ using the following definition: $A$ is an injective object in $\Bbb Z\text{-mod}$ iff the morphism Hom$_{\Bbb Z}(\Bbb Z,A) \...
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Self-injectivity of a trivial extension $B\ltimes Q/B$

Let $B$ be the ring of $2$-adic integers, and $Q$ be its field of fractions, and consider the trivial extension $R=B\ltimes (Q/B)$, that is, the set $B\times (Q/B)$ with coordinatewise addition and ...
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Writting objects in abelian category as limit of injective objects

In "EVERY MODULE IS AN INVERSE LIMIT OF INJECTIVES" (https://arxiv.org/pdf/1104.3173.pdf) it is proven that every modules is an inverse limit of injective modules. A natural question is ...
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Does category of finitely generated torsion $G$-modules has enough injectives?

Let $G$ be a profinite group. Then the category of discrete $G$-modules have enough injectives. Now I have a category of finitely generated and torsion $G$-modules with continuous $G$-action. Does ...
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Does $\operatorname{ker} \partial_n \to C_n$ being injective imply that $\operatorname{ker}\otimes G \to C_n\otimes G$ is injective?

Let $Z_n = \operatorname{ker} \partial_n$ and $B_n = \operatorname{im}\partial_{n+1}$. We know that the short exact sequence $$0\to Z_n \to C_n \to B_n \to 0$$ is exact. The surjectivity of $$g\colon ...
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$f:M \to Q$ a monomorphism with $Q$ injective and $g:M \to N$ an essential morphism, then there exists a mono $\bar{g}: N \to Q$ such $\bar{g} g= f$.

This one is based on (A) of the answer of this previous question For a projective cover $(\sigma, P)$ of a module $M$, $P$ is indecomposable implies $M$ is indecomposable Let $Q$ be an injective ...
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Why does the injective module have no projective summand?

Sorry if this is too elementary. The problem is from Auslander's Representation Theory of Artin Algebras, page 214 proposition 5.6. Let $\Sigma$ be an artin algebra(gl.dim$\Sigma$=dom.dim$_{\Sigma}\...
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Use of “$A$ is a domain” in the proof that $Q$ is an injective $A$-module iff it is divisible

Let $A$ be a PID. Then, an $A$-module $Q$ is injective iff $Q=rQ$ for every $r\neq 0$ in $A$. My question is, where is the property "A is a domain" used in the proof of the above? Can someone please ...
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Comparing the exactness of $(\prod_{\lambda\in \Lambda}M_{\lambda})$ and $M_{\lambda}$ on a long exact sequence.

Let $R$ be a ring with unity and $S=\text{End}_R(M), {}_{S}M_R$ be an $(S,R)$-bimodule and $\{M_\lambda\}_{\lambda\in \Lambda}$ be a family of modules. Define the map $\phi:M\to M,\phi(m)= mr,r\in R)$...
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$mod\Lambda$ does not have enough injectives

Let $\Lambda = \begin{pmatrix} \mathbb{Q} & 0\\ \mathbb{R}& \mathbb{R} \end{pmatrix}$, a subring of $M_2(\mathbb{R})$. Let $mod\Lambda$ be a category of finely generated $\Lambda$-modules. ...
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Are these the projective and injective representations of these quivers?

Find all indecomposable projective and indecomposable injective representations of these two quivers over the field $k$ up to isomorphism. I've drawn my answer in this picture. Can you please check ...
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Injective resolution of $K$ as a $K[x]/x^n$-module

I have read about the question here: Projective resolution of $k$ as $k[x]/(x^n)$-module?, which deals with a projective resolution of $K$ as $K[x]/x^n$-module, where $K$ is a field. And I am ...
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$\Bbb Z_n $ is an injective $\Bbb Z_n$-module

I am trying to prove or disprove that $\Bbb Z_n $ is an injective $\Bbb Z_n$-module, where $n$ is an integer $>1$. If $n$ is a prime, then $\Bbb Z_n$ is a field, so every $\Bbb Z_n$-module is ...
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Let $A$ be a PID, $M$ an injective finitely generated module. Prove that $M = 0$. [closed]

Help! Let $A$ be a PID, $M$ an injective finitely generated module. Prove that: $$ M = 0.$$
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Properties of injective modules

I am reading A course in Homological Algebra by Hilton and Stammbach. In the first chapter they showed that a $\Lambda$-module is projective iff it is a direct summand of a free module. They then ...
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Hartshorne's proof of Lemma III.3.3. (Localization of injective module is surjective)

I am reading Hartshorne's proof of Lemma 3.3 on Chapter III, that asserts that for every injective module $I$, the natural map $I\rightarrow I_f$ is surjective. The key of his proof is that for ...
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If $I$ injective then $\operatorname{Hom}_R(M,I)=0$ imply $M=0$?

Let $R=(R, \mathfrak{m},k)$ be a local ring with maximal ideal $\mathfrak{m}$ and residue field $k=R/\mathfrak{m}$. Let $I_k$ the injective hull of $k$, i.e., it's an essential extension of $k$ and an ...
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Existence of Projective Cover

Let $A$ be a finite dimensional algebra and let $M$ be a finite dimensional $A$-module. Then $M$ has a projective cover and I am trying to prove this. Let $\mathcal{C}$ be a module class and $X$ a ...
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Injectivity of the $\mathbb{Z}$ -module $\mathbb{Z}(p^\infty)$

Let $\mathbb{Z}(p^\infty)$, for a given prime number $p$, be the $\mathbb{Z}$-module with generators $\{a_{i}\}_{i=1}^\infty$ and relations $pa_1=0$ and $pa_{i+1}=a_i$. I would like to show that ...
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Why can't an embedding have self-intersections?

If I understood it correctly an embedding is an immersion $\phi:M \rightarrow N$ that is a homeomorphism and where $\phi (M) \subset N$. To which condition contradicts an immersion with a self-...
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whether the function is injective surjective or both

Let $f: \mathbb R^2\to\mathbb R$ be defined by $f(x,y)=x^2+y^3,$ then is it injective bijective or both? I try this: if $x_1^2+y_1^3=x_2^2+y_2^3$, it doesn't mean $x_2=x_1, y_2=y_1$. So it is not ...
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If (co)domain of map of chain complexes is in(pro)jective, then every quasi-isomorphism is a homotopy equivalence

I'm reading some lecture notes which are an introduction to homotopy theory, and there is a short section on chain complexes, where the difference between a quasi-isomorphism and homotopy equivalence ...
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Injective Linear transformation . [closed]

If the linear transformation $T:\mathbb R^3\to\mathbb R^3$ defined as $$ T(x,y,z)=(y+kz,x+ky,x-2y+z) $$ is injective, what is the value of $k$?
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Equivalent condition to exactness

Let $I$ be an injective $R$-module. Is it true that a short sequence of $R$-modules $$0\rightarrow A'\rightarrow A\rightarrow A''\rightarrow 0$$ is exact if and only if $$ 0\rightarrow \text{Hom}_R(A''...
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Is the delta module injective?

Is $M=\mathbb{C}[\partial_x,\partial_y]$ an injective $\mathbb{C}[x,y]$ module? The action is given by differentiation (swapping the role of $x$ and $\partial_x$). It is injective in the one variable ...
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Is $\mathbb{Q}_p/\mathbb{Z}_p$ an injective $\mathbb{Z}_p$-module?

I'm curious as to whether or not $\mathbb{Q}_p/\mathbb{Z}_p$ is an injective $\mathbb{Z}_p$-module. It's certainly an injective abelian group ($\mathbb{Z}$-module) since $\mathbb{Q}_p/\mathbb{Z}_p$ is ...
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Injective and quasi injective modules

Definition: An $R$-module $M$ is called quasi-injective module if for every submodule $N$, any $R$-homomorphism $N\to M$ extends to an endomorphism of $M$. How can I prove that every injective module ...
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Is direct sum of injective modules quasi-injective

Let $R$ be a ring with identity and $M$ be a injective $R$-module. Is $M^{(N)}$ (a direct sum of copies of $M$) a quasi-injective $R$-module? Here is my proof. Since M is injective, M is M-injective ...
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Why this s.e.s. splits?

Suppose that $$0\to(M/N)^+\to M^+\to N^+\to 0$$ is a short exact sequence of modules with $(M/N)^+$ and $N^+$ injective as defined here. Why $(M/N)^+\to M^+$ splits and what does it mean that it ...
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Example of a finite group for which $\mathbb Q$ is not injective.

I had to prove that the cohomology with rational coefficients of any finite group is trivial. I tried to prove it by proving that $\mathbb Q$ is an injective $G$-module, however it doesn't seem to me ...
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X is projective iff for injective module&arbitrary module

The problem is : Prove that an arbitrary R-module X is projective iff for every homomorphism $f : X \to B$ and every epimorphism $g : A \to B$ from an injective module A, there exists a ...
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Bimodule resolutions

Let A be a finite-dimensional algebra. Let M be a left A-module and N be a right A-module. Choose an injective resolution $E_1^*$ of $M$ in $A$-mod and an injective resolution $E_2^*$ of $N$ in mod-$A$...
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Proof verification: $R$ Noetherian implies direct limit of injective modules is injective

I'd like to verify that my proof that $R$ Noetherian implies that any direct limit of injective R-modules is injective. This proof works off of some ideas from this question and a paper by Clark, but ...
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Local Cohen-Macaulay ring over which every finitely generated module of finite injective dimension also has finite projective dimension

Let $(R,\mathfrak m,k)$ be a local Cohen-Macaulay ring. If every finitely generated $R$-module that has finite injective dimension also has finite projective dimension, then is it true that $R$ is ...
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Are Socle of a module and it's injective hull same?

If $E(M)$ is an injective hull of an $R$-$module$ $M$ then $Soc(M)=Soc(E( M))$. My attempt- abviously $Soc(M) \subset Soc(E(M))$ and since every essential submodule of $M$ is also an essential ...
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socle and essential extension of finitely generated mod over an artin algebra

In page 40 of Auslander‘s representation theory of artin algebra, the proposition 4.1, For $A$ in mod$\Lambda$ where $\Lambda$ is an artin algebra we have the following. a) $A=0$ iff $socA=0$ ...
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injective hull of $\mathbb{Z}_2$

I wonder what is the injective hull of $\mathbb{Z}_2$, what is its description? I believe that the injective hull of $\mathbb Z$ is $\mathbb Q.$
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On set of all $Z$-module homomorphisms as injective module

Let $\mathbb{Z}$ be the integers. The set $M$ of all $\mathbb{Z}$-module homomorphisms from a ring $R$ with unity to a divisible abelian group $A$ is known to be an injective left $R$-module. I am ...
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Is every submodule of an injective module injective?

I don't think this is the case but I can't find an error in my proof: Let $N \subset M$ be a submodule of an injective module $M$. Suppose we have maps $f:A \to N$ and $h:A \to B$ and we want to ...
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Equivalence in definition of quasi-Frobenius ring

A quasi-Frobenius ring is defined to be a ring which satisfies the following conditions (I'm phrasing it with left modules, but it's the same with right modules): (i) $R$ is Noetherian and self-...
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Proof of Baer's criterion: what set to apply Zorn on?

I'm reading the proof of Baer's criterion here: https://ncatlab.org/nlab/show/Baer%27s+criterion and I'm a little confused. Is it possible that they identify $M$ with its image in $N$? I think if ...
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Injective module characterisation

Is there an easy way to prove one of the following two implications? Every proof I saw uses some deeper embedding results or uses the functor approach ($Hom(R,.)$ exact functor) but I'm looking for ...
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injective dimension of nilradical multiplied by an injective module

I would like to prove the following: Let $R$ be a commutative Noetherian ring with identity, $I$ an injective module, and $N=\sqrt{(0)}$ (the nilradical). Then $I/NI$ is an injective $R/N$ module. ...
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Why does faithful module induce injection between Hom?

This is from a paper of Matsumura: The line that I don't understand is the faithfulness of $E$ implies injection $\text{Hom}(D,D) \rightarrow \text{Hom}(E,D)$. Why is this true? Thanks in advance.
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Left Artinian with non-trivial nilpotent ideals implies every left $R-$mod is injective

I know this is true, and couldn't recall the proof I'd learned, so I decided to try my hand at writing my own. Could anyone tell me if my work is correct, or give me some advice how to modify it. Or ...
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Projective and injective objects

I'm trying to prove the following: If $\mathcal{A}$ is an Abelian Category, then every object in $\mathcal{A}$ is projective if and only if every object is injective. My attempt: Since every object ...
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Over an integral domain is a module injective iff it is divisible?

Our professor gave us a proof for P.I.D.'s, but I believe I've found a proof that shows it works for any integral domain. If my claim is wrong, could someone explain why my proof fails? Or even just ...

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