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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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Enough injectives in the category of torsion abelian groups

Claim: The category of torsion abelian groups has enough injectives. I thought I had a proof of this, but discovered a mistake in my proof. I was trying to use the techniques of the usual proof that ...
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In R-Mod, all monomorphisms are equalizers

I want to prove that in the category of R-Modules, all monomorphisms are equalizers. We start by assuming that if $f : A \to B$ is mono and if $f$ equalizes $g : B \to C$ and $h : B \to C$ (i.e. $ g \...
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Theorem on Injective Dimension in Weibel's “An Introduction to Homological Algebra”

This fact is stated in Weibel's book without proof: If $A$ is a finitely generated module over a commutative Noetherian local ring $(R,\mathfrak{m},k)$, then $id(A) \le d$ is equivalent to $\...
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Let $R$ be an integral domain with a field of fractions $F$. Show that if $M$ is a $R$-module, then …

Let $R$ be an integral domain with a field of fractions $F$. Show that if $M$ is a $R$-module, then: $1)$ $M_F=F\otimes_R M$ is a divisible $R$-module $2)$ $x\rightarrow 1\otimes x$ is an essential ...
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A kind of horseshoe lemma on the injective resolution

We know that in an appropriate setting, with some exact sequence $0 \to A \to B \to C \to 0$ of $R$-modules with suitable $R$ (commutative ring with unity), we have injective resolution of $A,C$. Then,...
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Dimensional Vector spoace with injective surjective and bijective

I know there are similar questions on the internet, but I'm not getting it smart. i hope u can help me, Let $B,C$ finite dimensional Vector space with $\dim(C) = \dim(B)$ and $R \in Hom(B,C)$. Then ...
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Constructing injection into injective group

$\newcommand{\ZZ}{\mathbb{Z}}$ $\newcommand{\QQ}{\mathbb{Q}}$ $\newcommand{\Hom}{\mathrm{Hom}}$ Some time ago I tried to construct for a given abelian group $M$ functorially a group $I(M)$ which ...
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Embedding $\mathbb{Z}$-modules into injective $\mathbb{Z}$-modules

I want to show that any $\mathbb{Z}$-module $M$ can be embedded into an injective $\mathbb{Z}$-module (I'm in the process of showing this can be done for more general rings, but starting with this ...
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If P is a Projective R Module and Q is an Injective R module then $P \otimes_{R} Q$ is Injective

I am doing some basics on Protective, flat and Injective but I have no idea how to proceed for this one. Any help is appreciated!
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Is the (truncated) Witt ring an injective module over itself?

Let $k$ be a perfect field of characteristic $p$. I read that $W_n(k)$ is injective as a $W_n(k)$-module. I did not find a direct reference for this, but I assume it has to do with this question. My ...
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Proving this map is injective

Let $R$ be a ring and let $J$ be a left ideal of $R$. Let $s \in R$. We can make the module $R/J$. Let $\DeclareMathOperator{\ann}{ann}\ann_R(t + J) = {\{r \in R : rt + J = 0}\}$ be the annihilator. ...
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Reference requested: injective modules and subgroups

Let $G$ be a group, and let $H$ be a subgroup of $G$. Suppose that $E$ is an injective module over $\mathbb{Z}G$. Then I think that, as a consequence of Baer's criterion, $E$ is also an injective ...
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Enough injectives in the category of chain complexes

So I am a little confused about a question I thought would be obvious. Let $\mathcal{A}$ be an abelian category and let $\text{Ch}(\mathcal{A})$ be the category of chain complexes. Is it true that $\...
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Injective module

Let $R=M_n(D)$ the Matrix ring over a division ring and consider $R$ as a left module over itself. Is $R$ an injective module? I know that $R$ is free, hence it is projective. Is $R$ injective? I ...
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$\mathbb{Z}H$-projective modules in terms of projective $\mathbb{Z}G$-modules

Let $G$ be a group and $H\leq G$. I am trying to prove that silp$\mathbb{Z}H\leq$ silp$\mathbb{Z}G$ and spli$\mathbb{Z}H\leq$ spli$\mathbb{Z}G$, where silp$\mathbb{Z}G$ (resp. silp$\mathbb{Z}H$) is ...
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Quotient of injective module which is not injective

Let $R$ be commutative, and $V,W\in R$-$\textbf{Mod}$. As the title states, I am seeking an example of an injective $R$-module $V$ where $V/W$ is not injective. There is a question/answer here about ...
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Prove that no $\mathbb{Z}$-module is both projective and injective

Prove that no nontrivial unitary $\mathbb{Z}$-module(or equivalently, abelian group) is both projective and injective. My attempt: Since $\mathbb{Z}$ is a PID, injective modules are the same as ...
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If $Q$ is injective, then $\text{Hom}_{\mathbb Z}(R, Q)$ is injective.

I would like some hint for the following exercise: Suppose $Q$ is an injective $\mathbb Z$ right module. Let $R$ be a ring. Recall that $\text{Hom}_{\mathbb Z}(R, Q)$ has a $R$-right module ...
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Is R/S injective R-module?

In the book Exercises on Modules and Rings- Lam, Exercise 16.2 gives a ring as follows: K is a field, $\sigma\in End(K)$, $L=\sigma(K)$ such that $[K:L]=2$ (dimension is given as n in the book.) Let $...
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Let $R$ be a Noetherian Ring, then the category of $R$-modules has enough injectives

$R$ left-Noetherian, so all left ideals are finitely generated. I am trying to prove that the category of left $R$-modules has enough injectives. That is, given any left $R$ module $M$, there is an ...
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Element in a finitely generated torsion module on a PID with smallest non-zero annihilator

This question was motivated by this answer in order to get a simple proof of the structure theorem of finitely generated torsion modules over PID. Let $A$ be a PID and $M$ be a finitely generated ...
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$\mathbb{Q}_{\mathbb{Z}}$ is an injective hull of $\mathbb{Z}$

I want to prove that $\mathbb{Q}_{\mathbb{Z}}$ is an injective hull of $\mathbb{Z}$. Suppose that $H$ is an injective hull of $Z$. If I can show that $\mathbb{Q}$ contains an isomorphic copy of $H$, ...
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For $d\mid m$, $\mathbb{Z}/d\mathbb{Z}$ is not an injective $\mathbb{Z}/m\mathbb{Z}$-module when some prime divides $d$ and $\frac{m}{d}$

Suppose $d\mid m$. Show that $\mathbb{Z}/d\mathbb{Z}$ is not an injective $\mathbb{Z}/m\mathbb{Z}$-module when $\exists p$ prime with $p\mid d$ and $p\mid \frac{m}{d}$. $\textbf{My attempt:}$ We ...
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Injective Linear Transformation $K[x]_{\leq 4}\rightarrow V$

When $V$ is the vector space of all $2\times2$ matrices, why is no injective linear transformation $T: K[x]_{\leq 4}\longrightarrow V$? ($K$ is a field)
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Injective object in the category of projective systems of $R$-modules.

I am trying to prove that the category of projective systems of $R$-modules indexed over a directed poset $I$ has enough injective objects. As suggested in Jensen's book ("Les foncteurs dérivés de ...
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Example of reduced module

An $R$-module $M$ is a reduced module if $M$ has no nonzero injective submodules. I need example of a reduced module.
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Problem based on Projective and Injective Module

Show that the following are equivalent for a ring: (1) any $R$-module is projective. (2) any $R$-module is injective This is problem in Dummit Foote book, problem number 6, page 403. I was trying ...
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Decomposition of injective modules over polynomial rings

Let $A=\mathbb{C}[x_1,\ldots,x_n]$ be a polynomial algebra over the complex numbers. I am interested in injective modules over $A$. Since $A$ is projective over itself, the $\mathbb{C}$-dual module $...
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injective hull of a ring that is not integral domain

If $R$ is an integral domain, then the injective hull of $R$ ($E(R)$ in symbol) is $Q(R)$. But what is $E(R)$, where $R$ is not an integral domain? In particular what is $E(R)$, where $R=k[x,y]/(x^2,...
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direct sum of injective hull of two modules is equal to the injective hull of direct sum of those modules

why is the direct sum of injective hull of two modules equal to the injective hull of direct sum of those modules? In other words, $E(M\oplus N)=E(M)\oplus E(N)$
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Is zero a non-divisible $\mathbb Z$-module?

I want to find two injective $\mathbb Z$-modules for which the tensor product is not injective, so i used to try $\mathbb Q/\mathbb Z\otimes_{\mathbb Z}\mathbb Q/\mathbb Z$, which is equal to zero, so....
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Tensor product of two injective R-modules is not necessarily injective

Would you please help me to find an example of two injective R-modules (R is a commutative ring) for which the tensor product is not injective?
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injective hull as the “topological dual to the completion”

Dmitry Kaledin's notes (Russian) on cohomology of sheaves on algebraic varieties define (lecture 14) the injective hull of the residue field $k$ of a local ring $R$ as an injective module $I$ such ...
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Without using Baer’s criterion show $\Bbb{Z}/n\Bbb{Z}$ is self-injective

Unfortunately I have only seen examples of this using Baer’s criterion. I am following Wiebel’s an introduction to Homological Algebra as a text. I was given a hint: Show that a morphism $f: A \to ...
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Finding an injective hull

Let $R=\mathbb{Z}_4$ and consider the right $R$-module $E_R=\mathbb{Z}_4\oplus\mathbb{Z}_4$ and the submodule $M=\{(\bar{0},\bar{0}),(\bar{2},\bar{2})\}$. Identify two distinct injective hulls of $M$ ...
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When is a quotient of an injective module injective?

Let $M$ be a module and $I$ an injective module. We know that $I/M$ is not injective in general. But can we assume some conditions on $M$ such that $I/M$ is injective? Thank you!
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Direct sum of injective modules

Let $M$, $(E_i)_{i \in I}$ be $A$-modules. 1) There exists a canonical injective morphism $\varphi :\bigoplus_{i \in I}$ Hom$(M,E_i) \to $ Hom$(M, \bigoplus_{i \in I}E_i)$ 2) If $M$ is finitely ...
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Non-quasi-coherent injective sheaves

Hartshorne in his Residues and Duality shows that for locally noetherian (pre)schemes $X$, the injective objects in the category of $\mathcal O_X$-modules decompose (uniquely) into a direct sum of ...
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Modules and homomorphism on a domain

Let $R$ be a domain. 1) If $M$ is a non-zero injective $R$-module, then every non-zero homomorphism from $M$ to $R$ is surjective. 2) If $M$ is a non-zero projective $R$-module, then there exists a ...
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225 views

Proof of the comparison theorem (homological algebra) for injective resolutions

I am trying to prove the existence part of the comparison theorem for injective resolutions. Every reference I find says that it is just the dualization of the proof for projective resolutions, but I ...
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67 views

Quotient by annihilator of an element is an injective module

Let $R$ be a commutative ring, $M$ be an $R$ module and $m\in M$. I am trying to prove or disprove that $M/ann(m)$ is an injective $R$ module. Any suggestions are welcome. I am expecting the answer ...
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How do the minimal right almost split maps ending in an injective module of a path algebra look like?

Let $Q$ be a finite quiver without oriented cycles and let $k$ be a field. Then $A = kQ$ is a finite-dimensional path algebra. Let $e_1, \dots, e_n$ be the idempotents corresponding to the vertices $1,...
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On cyclic right $R/I$-modules for two sided ideal $I$ of $R$

Let $R$ be a ring such that every cyclic right $R$ module is either injective or projective and let $I$ be a two sided ideal of $R$ . Then why is every cyclic right $R/I$-module either injective or ...
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Module and localization of injective

Let $K$ be a field and let $I$ be an infinite set. We put $R = K^I$ (copies of $K$), $J = K^{(I)}$ and $S = \{1 − r \mid r ∈ J\}$. Then $R/J\simeq S^{−1}R$, $R$ is an injective module, but $R/J$ is ...
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The set$\quad${${E(R/I)|I\leq R}$}$\quad$contains an isomorphic copy of each indecomposable injective.

I am reading Rings and Categories of Modules by Anderson and Fuller. In the book page 293: An indecomposable injective left $R$-module must be the injective envelope of each of its non-zero ...
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Is every vector space an injective module?

Let R be a commutative ring and $Q$ an $R$-module, we say that $Q$ is injective if it has the property that for all $R$-modules $M$ and $N$ and homomorphisms $$f\ :\ M\to Q$$ and $$\phi\ :\ M\to N$$ ...
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How to get $fR \otimes_R-$ sends injective modules to injective modules?

Let $R$ be any ring. $f$ is an idempotent of $R$ such that $fR$ is a faithful right module. I have seen that "Since $fR_R$ is projective, the functor $fR \otimes_R-$ sends injective $R$-modules to ...
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Multivariable polynomial ring injective module? [closed]

Is the polynomial ring $\mathbb{R}[x_1, \ldots ,x_d]$ an injective module over itself?
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A module which is an extension of a regular and a preinjective module is regular?

In proposition 4.4 of the paper "Representations of Wild Quivers" by Otto Kerner the setting is the following: $A$ is a wild hereditary path algebra $E$ is an indecomposable regular module He proves ...
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Using Baer's criterion for abelian groups.

I am currently working through the chapters on injectivity in Donald Passman's A Course in Ring Theory, and have run into conceptual difficulty while attempting one of the exercises. I am trying to ...