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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Do the construction of the injective sheaves always lead to the contractible ones?

For a presheaf $F:Sm/k\rightarrow Groups$, let's define the presheaf $CF(X):=F(X\times \mathbb{A}^1)$. There are two natural pullback morphisms from $CF$ to $F$, denoted by $i_0^*$ and $i_1^*$. These ...
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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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Infinite periodic injective resolution of $\mathbb{Z}/(p)$

Consider $\mathbb{Z}/(p)$ as a $\mathbb{Z}/(n)$-module for $p$ prime. I'm asked to show that there exists an infinite injective resolution of $\mathbb{Z}/(p)$. Since $\mathbb{Z}/(n)$ is injective over ...
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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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Essential Extensions and Torsion Subgroups

I'm working on an Exercise in Hilton/Stammbach's A Course in Homological Algebra that deals with essential extensions and torsion subgroups of abelian groups. So far I've shown the following: If we ...
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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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Show that $\mathbb{Q}/\mathbb{Z}$ is an injective module and that it is not a projective module

So this is what I have for the injectivity: Knowing $\mathbb{Z}$ is a principal domain, $\mathbb{Q}/\mathbb{Z}$ is an injective module if and only if $\mathbb{Q}/\mathbb{Z}$ is divisible. Checking the ...
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On the generators of the injective hull of the residue field of $\mathbb C[x,y]/(x^2, xy,y^2)$

Consider the Artinian local ring $R:=\mathbb C[x,y]/(x^2, xy, y^2)$ with residue field $\mathbb C$. Let $E:=E_R(\mathbb C)$ be the injective hull (as an $R$-module) of the residue field (also known as ...
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Why isn't this statement about direct sums of injective modules contradicting the proposition?

I am trying to read Tsit-Yuen Lam's "Lectures on Modules and Rings" in order to understand why any abelian group can be injected in an injective abelian group, but I got confused by the ...
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Injective Modules Motivation & Intuition

A module $M$ over a commutative ring $R$ is called a 'injective module' if it satisfies certain universal property explaned here. Question: Is there any intuition how to think concretely about ...
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Geometric Intuition of Baer Criterion

The critical step in the proof of Baer's Criterion involves defining a map $g(r)=f(rx)$ on the constructed ideal. I think this bears a similarity with extending a map from $\mathbb R^n$ to $\mathbb R^{...
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Why is $\operatorname{Fun}(A^{*},\mathbb C^{*})$ injective?

Everything below is a $\mathbb Z-$mod. Let $A^{*}$ = $(\operatorname{Hom}A,\mathbb C^{*})$. I(A) = $\operatorname{Fun}(A^{*},\mathbb C^{*})$. [note : only functions,not homomorphisms] Why is I(A) an ...
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Proof regarding quotient sets and functions.

Okay so I was kind of winging it, especially in the second direction. I wanted to make sure what I wrote is not complete nonsense.. Assume $f$ is injective. Since $R$ is an equivalent relation, it ...
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Question on injective envelopes

Let $A$ be an Artin algebra and let $\text{mod}(A)$ denote the category of finitely generated left $A$-modules. Let $S$ be a simple module in $\text{mod}(A)$ and let $\iota_S: S \rightarrow I(S)$ be ...
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An Exercise on projective covers and injective envelopes of simple modules over an Artin algebra

Let $A$ be an artin algebra and let $\text{mod}(A)$ denote the category of finitely generated left $A$-modules. Let $S$ be a simple module in $\text{mod}(A)$. Let $\iota_S: S \rightarrow I(S)$ and $\...
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Surjection of $\text{Hom}(-,I)$ where $I$ is injective

Let $I$ be an injective abelian group and $G\to H$ an injective morphism of groups, not necessarly abelian. If $G,H$ were abelian, then we would have a surjection $$\text{Hom}(H,I)\to \text{Hom}(G,I).$...
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Non trivial homomorphism into injective envelope of a simple composition factor

Let $A$ be an artin algebra and let mod$(A)$ be the category of finitely generated $A$-modules. Let $S$ be a simple module and let $M \in$ mod$(A)$. We further assume that $S$ is a composition factor ...
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Why is local cohomology $R$-linear functor?

Let $ R = k[x_1, \ldots, x_n] $ and $\mathfrak{m} = (x_1, \ldots, x_n)$. Let $N$ be an $R$-module, then the $\mathfrak{m}$-torsion functor is defined as $\Gamma_{\mathfrak{m}}(M) = \{ m \in M \text{ s....
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Prove that every abelian group is a subgroup of divisible group.

I need a proof that every abelian group is a subgroup of divisible group (to make sure that every object of the category of $\mathbb Z$-modules has injective resolution). I found a proof on group ...
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Surjectivity of character module map implies injective module map?

I'm trying to understand modules. And I want to know, by definition of a module $M$, the underlying group structure on $M$ is abelian. Does that mean that every $R$ module $M$ is also a $\mathbb{Z}$ ...
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Confusion in module structure of indecomposable injectives over quiver path algebra

I got very confused while working out the explicit module structure on the indecomposable injective modules $I_i$ over a finite-dimensional quiver path algebra $kQ/I$. I know that the right injective ...
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Why can $M$ be an abelian group?

In Rotman's, "An introduction to homological algebra", there is a proof of that the categories of modules has enough injectives. In the proof, he proves that there is an injective map from $...
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Why is $R/I$ cyclic?

The question is in the title, in Rotman's "An introduction to homological algebra", page 125 is the next proposition: After the diagram, it says that $R/I$ is cyclic, why ?
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Tensor product of injective modules

Searching for some properties of injective modules I have come across this question: Is it true that Tensor product of injective modules is injective? Unfortunately in that question, there are no ...
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Is restriction of scalars preserving injective modules equivalent to flatness?

Given any ring homomorphism $R \to S$, if $S$ is a flat right $R$-module, then any injective left $S$-module is also injective as a left $R$-module. Now, I'm wondering whether the converse is true. ...
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Computing the number of projective-injective modules

Let $A$ be a finite dimensional $\mathbb{K}$-algebra, where $\mathbb{K}$ is an algebraically closed field. How does one compute (homologically) the number of projective-injective $A$-modules? Maybe ...
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Injectivity or projectivity criterion in modules and model categories

Here in Hovey's book on Model Categories in as opposed to errata there is already injective in the lemma 2.2.8. The errata says injective and not projective. Also I wonder what is $A$ in $A\to Q$ ...
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Prove that $f_a:M\to M,m\mapsto ma$ is an isomorphism whenever it injective iff there is nonzero linear map $f:Ε\to Μ$ with $Ε$ injective.

Prove that the following are equivalent for a commutative noetherian ring with identity: (a) For every nonzero $R$-module $Μ$ there is a nonzero linear map $f:Ε\to Μ$ with $Ε$ injective. (b) For every ...
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Baer's criterion for injective modules

In Dummit and Foote section 10.5, proposition 36 says Let $R$ be a PID. $Q$ be a $R$ module. $Q$ is injective if for all $r\in R$, $rQ=Q.$ $\cfrac{\mathbb{Z}}{4\mathbb Z}$ is injective as $\cfrac{\...
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If $G=\bigoplus_i\Bbb Z/p_i^{n_i}\Bbb Z$, then for which integers $r\neq\pm 1$ is $G=Gr$?

Let $G=\bigoplus_i\Bbb Z/p_i^{n_i}\Bbb Z$ be a finite Abelian group with $p$ prime, $n,i$ integers. For a fixed $r\in \Bbb Z$, let $f_r$ be a multiplication endomorphism on $G$ for which $f_r(g)=gr$ ...
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Proof that Divisible Modules are Injective over a PID

I'm reading the proof given in Hilton/Stammbach's homological algebra book that over a PID, injective modules and divisible modules are the same. (Thm. 7.1 in chapter 1, pp. 31-32). I'm stuck on a ...
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T.Y. Lam's divisible module definition and factor modules.

In surveying LMR of T.Y.Lam and get the divisible module (for any ring with unity not necessary an integral domain) definition as follows: ``A right $R$-module $I_R$ is called divisible if and only if ...
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The dual for Abelian group on Q/Z

Let $A$ be any Abelian group. Take $A^\star:=\operatorname{Hom}(A,\Bbb{Q}/\Bbb{Z})$ be the dual of $A$. Then is $A=0$ equivalent to $A^\star=0$?
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$\Lambda/\mathfrak{r}\cong \operatorname{soc}(\Lambda)$ as a criterion for self-injectivity

I've been working through the exercises of Auslander, Reiten, and Smalø's Representation Theory of Artin Algebras, and have gotten stuck on their Exercise 4.12, which asks: Let $\Lambda$ be an Artin ...
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Why injectivity class doesn't imply weakly reflective

In the snippet below (taken from Rosicky, Adamek: On injectivity in locally presentable categories) in the context of locally presentable categories it seems that injectivity class doesn't imply ...
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Quasi injective direct sum impy injectivity.

On his book, "Algebra: Rings, Modules and Categories", C. Faith gives the next assertion: $$\mbox{If} 𝐴\oplus𝐵\mbox{ is a }𝑄𝐼\mbox{-}module\mbox{ thus }𝐴\mbox{ and }𝐵\mbox{ are ...
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Does $M$-injectivity impy injectivity?

Let $N,M\in R$-mod. We say that $N$ is $M$-injective if for any $L$ submodule of $M$ and any homomorphism $f:L\rightarrow N$ exists a homomorphism $g\colon M\rightarrow N$ that extends $f$. In the ...
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How to verify this property of homomorphism

While studying a result( Lemma 3.11) on page 196 of Thomas Hunger Ford I have a question whose image I am adding. Image: Question: I am not able to prove g(ab)= g(a) g(b) property of group ...
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A question in verifying conditions of Zorn' s Lemma

I am studying algebra from Thomas Hunger Ford and I have a question in a thoerem on page $194-195$ of Chapter Modules. I am adding it's image. Edit :Question : How can I prove that every chain has a ...
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Quasi Injective Ring equivalence.

Let $R$ a ring (non conmutative in general) with unit. We say that $R$ is a $QI$-Ring (quasi-injective ring) iff for every $R$-mod $Q$, QI (quasi-injective module) is also injective. I need prove the ...
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Projective and injective modules are compatible with block decompositions of algebras in an obvious way.

Projective and injective modules are compatible with block decompositions of algebras in an obvious way. $A$ is a $k$-algebra and $b$ is an idempotent in $Z(A)$. (i)Let $P$ be a projective(resp. ...

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