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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Reference for Minimal Injective Resolution

I've been loking for reference and the definition of minimal injective resolution. I've found the wiki entry of Nakayama Conjeture. But I have problem undestanding what a minimal injective resolution ...
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What on Earth does Lang mean by "write the second square in the form..." in proof of Lemma 5.2, Homotopies of Morphiphsms of Complexes?

On the top of page 789 (I own the hardcopy of the book btw ;) it says: Next we must construct $f_1$. We write the second square in the form $$ \require{AMScd} \begin{CD} 0 @>>> E^0/M @>&...
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Extending module homomorphisms into an injective cogenerator

I have a discrete valuation ring $A$, an infinite collection $(M_i)_{i\in I}$ of non-zero $A$-modules and an injective co-generator $\Theta$ for the category $\text{Mod}_A$ of $A$-modules. Also set $M:...
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Relative injective modules

Let $R$ be a ring with unity. Let $A$ and $B$ be any right $R$-modules. Recall that $A$ is said to be $B$-injective if every homomorphism $f:B'\to A$, where $B'$ is a submodule of $B$, can be extended ...
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does the flat pre-cover need to exist for a left module?

All modules do have the flat cover by a result from the year 2001. What can be a ring $R$ and a left $R$-Module $M$ which doesn't have (a/the) flat pre-cover ? How can I construct such $R$ and left ...
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example of injective dimension of finitely generated module

Let $R$ be a commutative local ring. $M$ and $N$ are two finitely generated $R$-module of finite injective dimension. I want to fine an example of $M$ and $N$ such that $injdim(M)\neq injdim(N)$ Does ...
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Submodule of injective modules is injective

I was asked to prove that, given $A$ a ring, it is equivalent to say that "every submodule of a projective module is projective" and that "every submodule of an injective module is ...
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Is direct limit of injective étale sheaves injective?

Is the following statement true? if so where can I find a proof for reference purposes? Direct limit of injective étale sheaves of abelian groups on a Noetherian scheme is injective. This goes back ...
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Meaning of 'set of well-ordered sequences'

I'm trying to make sense of a construction of a module given in the following research paper: A New Construction of the Injective Hull, Fleischer, 1968. On the second page, a module $F$ is constructed,...
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If there exist monomorphisms between injective modules, they are isomorphic.

Suppose that $X_R$ and $Y_R$ are injective and $\theta:X\to Y$ and $\phi:Y\to X$ are monomorphisms. I need to prove that $X\cong Y$. My trial. Set $K=\theta(X)\leq Y$. Since $X$ is injective and $\...
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when are graded injective modules graded and injective?

Define a graded injective module over a graded ring $R$ to be an injective object in $GrMod-R$ (the category of right graded $R$-modules). From the little research I have done, a graded injective ...
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If every homomorphic image of an injective module is also injective, then every submodule of a projective module is projective

Let $R$ be a ring with $1$. All modules considered in this problem are unitary right $R$-module. Assume that every homomorphic image of an injective module is also injective. I need to prove that ...
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Equivalent Definition of injective $R$-module using exact sequences [duplicate]

We call an $R$-module $D$ injective if one of the two equivalent conditions holds: If $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ is an exact sequence of $R$-modules then the induced ...
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Importance of (category of) injective objects

I understand that injective objects play a central role for doing homological algebra in an abelian category, for example having enough injectives allows one to form injective resolutions which can be ...
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When $\mathrm{Hom}$ functor commutes with colimits in a category of modules? [duplicate]

I was looking through this question, and there's a thing I don't understand why it holds. I mean the next statement in the answer by @Peter McNamara: Since $R$ is Noetherian, $I$ is finitely ...
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Use injectivity of $\Bbb Z/n\Bbb Z$ over $\Bbb Z/n\Bbb Z$ to prove Prüfer's First Theorem

I'm working on the part (b) of the exercise at P.67 of Brown's Cohomology of Groups, which is stated as follows: Let $R = \mathbb{Z}/n\mathbb{Z}$. Show that $R$ is an injective $R$-module. Deduce: (a)...
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Injective projective modules over left artinian ring

When I read Rings and Categories of Modules written by Frank W.Anderson and Kent R.Fuller, I can't understand the proof of Theorem 31.3 (Page 338) Let $R$ be a left or right artinian ring with $J=J(R)...
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$\mathbb Q/\mathbb Z$ and $\mathbb R/\mathbb Z$ are injective cogenerators for $\mathbb{Z}\textbf{-Mod}$

I wish to show that The abelian groups $\mathbb Q/\mathbb Z$ and $\mathbb R/\mathbb Z$ are injective cogenerators for the category $\mathbb{Z}\textbf{-Mod}$. Recall that An object $E$ in a category ...
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Direct limit of n-presented modules?

A module $M$ is said to be $n$-presented If there exist an exact sequence $$F_{n}\to F_{n-1}\to \cdots \to F_{1}\to F_{0}\to M$$ with each $F_{i}$ is free finitely generated. For example $M$ is $0$-...
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What conditions are necessary for an Artinian principal ideal ring to be uniserial?

Let $R$ an artinian commutative ring such that every factor ring (including $R$) is QF (quasi-frobenius, i.e. noetherian and self-injective). I'm looking for conditions to reach that this is a ...
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Some conclusions with respect to character module $M^{\ast}=Hom_{\mathbb{Z}}(M, \mathbb{Q/Z})$

Given an $R$-module $M$, we define its character module as $M^{\ast}=Hom_{\mathbb{Z}}(M, \mathbb{Q/Z})$. I have already proved that $M$ is a flat module iff $M^{\ast}$ is an injective module. Now I am ...
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The functor $\operatorname{Hom}(-,M)$, is it faithful?

Let $M\neq 0$ be a left $R$-module, I want to prove that if $M$ is injective then for every left $R$-module $A$, $\operatorname{Hom}(A,M)=0$ implies $A=0$. Indeed by contradiction if $A$ contains a ...
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Injective modules and prime ideals

Let $R$ be a ring with $1$. We say that a right $R$-module $M$ is indecomposable if $M$ can not be decomposed as the internal direct sum $M=L \oplus K$ with $L$ and $K$ non-zero. Suppose that $R$ is a ...
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Abstract Nonsense vs. explicit definition of first lie algebra cohomology $H^1(\mathfrak g, E)$

Context Many texts introducing a cohomology functor seem to take two approaches: Saying „take the derived functors of this half-exact functor“ without doing anything explicit or Defining $Z, B$ ...
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Does Repf($G$) have enough injectives?

Let $G$ be a group scheme over a field $k$. My question is Does Repf(G), the category of finite-dimensional linear representations of $G$, have enough injectives? It is well-known that Rep(G), the ...
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Prove that if $M=f(M)+\ker(f)$ and $S=\text{End}_R(M)$ is a reduced ring, then $S$ is regular

Consider a right $R$-module $M$ with a reduced endomorphism ring $S=\text{End}_R(M)$. If $M=f(M)+\ker(f)$ for every $f\in S$, then prove that $S$ is a regular ring. I tried to prove that in the ...
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Do images and kernels of invariant modules have the form $Mr$ and $\text{Ann}(r)$ respectively for some $r\in R$?

Let $M$ be an $R$-module and $N$ a submodule of $M$. $N$ is said to be a fully invariant submodule of $M$ if $f(N)\subseteq N$ for all $f\in \text{End}_R(M)$. We call an $R$-module $M$ invariant if ...
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Meaning of injectives objects in a category

I'm struggling to understand the meaning/motivation behind injective objects in (abelian) categories, especially in the context of group cohomology. They seem to be mostly mysterious as one mostly ...
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Coproduct of injective modules

I read many papers that looked for sufficient conditions for coproduct of injective modules to be injective modules, but I didn't find the reason why they always look for this. I want to know the ...
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How can i show intersection of injective modules?

Let $E=\mathbb{Z}_4\oplus\mathbb{Z}_4$, regarded as a module over $\mathbb{Z}_4$. Let $M= \{(0,0),(2,2)\}\subseteq E$ be a submodule. Then how can I show that $M$ is the intersection of $2$ injective ...
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When is $(M) / (M ⊗ I)$ $\simeq$ to $M/(MI)$?

Let $N$ be a R-Module isomorphic to $R/ker(\pi)$ with $\pi: R \rightarrow N$ the surjection that associate to each element of N an element of the canonical basis of R. One has the following exact ...
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Injective objects in the category of finite groups.

I have to describe the injective objects of the category of finite groups. I see that 1 is an injective object, but are there any others? If yes why, if no why not?
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Every ideal is projective if and only if every divisible $R$-module is injective.

For an integral domain $R$ show that: Every ideal is projective if and only if every divisible $R$-module is injective. I read this question in a book, but I don't know where I should start to prove ...
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Find indecomposable projective/injective module over $k[x]/ \langle x^n \rangle$

Problem: find indecomposable injective modules and indecomposable projective modules over $k[x]/ \langle x^n \rangle. (n\geq 2)$. From this post, I learned that any indecomposable module over $k[x]/ \...
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$M$ is a projective module iff $D(M)$ is an injective module

Assume that $A$ is a finite dimensional $k$-algebra, and $M$ is a left $A$-module. Let $D=\operatorname{Hom}_k(-,k)$. Problem: $M$ is a projective module iff $D(M)$ is an injective right $A$-module. ...
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Change of variable for a bijective function

If I have a function $h:[0,r] \rightarrow [0,m]$, where $r$, $m>0$. Assume I have the following integral: $$\int_0^m \frac{dt}{g(t)^\frac{1}{p}},$$ where $g$ is a given function, and $p>1$. My ...
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Vector Space over a Division Ring [duplicate]

Prove that every vector space over a division ring D is both a projective and an injective D-module. This question is from an assignment in Module Theory. I am familiar with the definitions and are ...
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Top-injective and indiscrete

I have found one proposition in some papers about pure injectivity topological modules. Proposition: A topological space is categorically Top-injective if and only if it is indiscrete. Proof: $(\...
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Injective modules problem

Let $R$ be a ring. Suppose that $X$ and $Y$ are left injective $R$-modules and $\theta: X \to Y$ and $\phi:Y \to X$ are $R$-monomorphisms. I want to prove that $X \cong Y$. This is what I have done so ...
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Clarification about the Injective co-generator $\mathbb{Q/Z}$

Context : I am currently self-studying homological algebra and would like to garner greater intuition about injective modules/resolutions; in particular about how injective co-generators "work&...
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Is the statement that only one of $\mathbb{Q}$ and $\mathbb{R}$ is an injective abelian group consistent with $ZF\neg C$?

Is the statement that only one of $\mathbb{Q}$ and $\mathbb{R}$ is an injective abelian group consistent with $ZF\neg C$ (assuming, of course, that $ZF$ is itself consistent)? Note that if $\mathbb{Q}$...
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Hom group $Hom_{\mathbb{Z}/m}(I, \mathbb{Z}/m)$ of an Ideal $I$

I'm trying to show that $R= \mathbb{Z}/m$ is an injective $R$-module using Baer's criterion (Weibel, Homological Algebra, exercise 2.3.1). I want to find the homomorphism module $Hom_R(I, R)$, then ...
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A One-Dimensional Local Ring Admitting a Finitely Generated Reflexive Module with Finite Injective Dimension Must Be Gorenstein

Let $(R, \mathfrak m)$ be a one-dimensional commutative Noetherian local ring. Let $M$ be a finitely generated $R$-module with finite injective dimension $\operatorname{injdim}_R(M).$ One can prove ...
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If every ideal is a direct summand, then every module is injective

I am trying to prove the following result: Let $R$ be an algebra. Show that every $R$-module is injective if, and only if, every ideal is a direct summand of $R$. The $(\Rightarrow)$ implication is ...
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Show that $\operatorname{Hom}(Ae_{1,1}, Ae_{2,2}) = 0$

Question If $K$ is a field, fix $A = \left \{ \begin{pmatrix} a & 0\\ b & c \end{pmatrix} : a,b,c \in K \right \}$ and $e_{1,1} = \begin{pmatrix} 1 & 0\\ 0 & 0 \end{pmatrix}$, $e_{2,...
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Show that $R$ is self-injective iff every finitely generated projective right $R$-module is injective

While studying for an upcoming exam, I've crossed the following problem: A ring $R$ is self-injective if, and only if, every finitely generated projective right $R$-module is an injective right $R$-...
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An injectivity criterion for $\mathbb Z/m\mathbb Z$ as $\mathbb Z/m\mathbb Z$ module

Let $R=\mathbb Z/m\mathbb Z$ where $m>1$. Show that $R$ is injective as an $R$-module iff for any abelian group $A$ such that $mA=0$ and for any subgroup generated by an element $a\in A$ of order $...
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$\pi^* : \Omega^r(P_n(R)) \rightarrow \Omega^r(R^{n+1}\setminus \{0\})$ is injective

Let $\pi : R^{n+1} \setminus \{0\} \rightarrow P_n(R)$ be the canonical projection. Show that the induced map $$\pi^* : \Omega^r(P_n(R)) \rightarrow \Omega^r(R^{n+1}\setminus \{0\})$$ is injective ...
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Over a ring without unity: A module is injective iff every certain exact sequence splits.

In this question let us assume that $R$ is a ring without unity. Here are two conditions for an $R$-module $J$. (a). $J$ is injective, that is, given any $R$-module monomorphism $g:A\to B$ and a ...
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Proving a Certain Subgroup of $\mathbb{Q}/\mathbb{Z}$ is Divisible

In an effort to compute the injective envelope of $\mathbb{Z}_p$ for $p$ prime, I need to show that the group $A \subset \mathbb{Q}/\mathbb{Z}$ generated by $\{1/p^r : r \in \mathbb{Z}^+\}$ is ...
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