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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When $f\colon\mathbb{Z}^{+} \to\mathbb{R}$ satisfys $f(2x)=2f(x)$, then is it injective? [closed]

If $f\colon\mathbb{Z}^{+} \to\mathbb{R}$ satisfys $f(2x) = 2f(x)$. Is it injective? I’m new in analysis and can’t understand how to approach this problem. Should I find an analytic expression for $f(x)...
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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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Lemma 8, Section 4.3 of Hungerford’s Algebra

Let $R$ be a ring with identity. A unitary $R$-module $J$ is injective if and only if for every left ideal $L$ of $R$, any $R$-module homomorphism $L\to J$ may be extended to an R-module homomorphism $...
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Left exact functors preserve injective objects

Let $G$ be a group and let $\operatorname{\textbf{Mod}}_G$ be the abelian category of $G$-modules, which has enough injectives. Let $A\in \operatorname{Mod}_G$ and let $(-)^G$ the functor \begin{...
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Does an exact sequence of holonomic D-modules imply an exact sequence in the solution space?

Here is a probably quite basic question on holonomic D-modules, but I am only a physicist so please bear with me. If I have the weyl algebra $D$ in $n$ variables, and an exact sequence of holonomic $D$...
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The injective hull of the ring of all eventually constant sequences of elements of $\mathbb{Z}_2$

Let $R$ be the ring of all eventually constant sequences $(x_n)_{n\in \mathbb{N}}$ of elements of $\mathbb{Z}_2$. It's known that the injective hull $E(R_R)$ of $R_R$ is $S:=\prod_{n\in \mathbb{N}} \...
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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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Injectivity of special orthogonal group

Let $GL_{n}(\mathbb{R})=\{A_{n\times n}\mid A \text{ invertible matrice}\}$, $SL_{n}(\mathbb{R})=\{A\in GL_{n}(\mathbb{R})\mid det(A)=1\}$ be a special linear group and $SO(n)=\{A\in O(n)\mid det(A)=1\...
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How Can We Prove Flatness from an Induced Exact Sequence?

I want to prove that for every short exact sequence $$ O \to A \xrightarrow{f} B \xrightarrow{g} C \to O $$ of $R$-module homomorphisms, if the induced sequence $$ O \to M\otimes_R A \xrightarrow{\...
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If a Short Exact Sequence is Split-Exact, Does that Mean it is Left-Split?

If a short exact sequence is split-exact, can we conclude it is left-split? Motivation: I am asking this question because I want to prove that if every $R$-module is projective, then every $R$-module ...
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How Should I Prove this Function is Well-defined?

I am trying to prove that the dual sum of P$_1$ and P$_2$ is projective iff P$_1$ and P$_2$ are projective. I am done with every aspect of the proof. I have this diagram which commutes: Now, let h: P$...
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Is this a valid proof of Well-definedness?

We are given $L, M$ and $N$ as unitary $R$-modules, and $f:M\rightarrow N$ as an isomorphism. We need to prove that $f^*:Hom_R(N,L)\rightarrow Hom_R(M,L)$ is an isomorphism. I started by defining the ...
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How Should I Prove this Function is an Isomorphism?

I have this question: Let $L$, $M$ and $N$ be unitary $R$-modules. Let $f: M \to N$ be an $R$-module isomorphism. Prove that the map $f_*: \text{Hom}_R(N,L) \to \text{Hom}_R(M,L)$ is an isomorphism. ...
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A module $J$ is injective iff every short exact sequence of the form $0\to J\to A\to B\to 0$ splits. [duplicate]

A module $J$ is injective iff every short exact sequence of the form $0\to J\to A\to B\to 0$ splits. I have seen these similar questions 1, 2, 3, but none contain a proof of this statement above. Here ...
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Question about proof that two modules are injectively equivalent.

Assume we have two exact sequences $$0 \to M \xrightarrow{\varphi} Q \xrightarrow{\psi} \mathcal{L} \to 0$$ and $$0 \to M \xrightarrow{\varphi'} Q' \xrightarrow{\psi'} \mathcal{L}' \to 0$$ where $M,Q,...
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Is $k[x, x^{-1}]$ a (graded) injective $k[x]$-module

Consider $k[x]$ with the usual grading, and the graded $k[x]$-module $k[x, x^{-1}]$. Is it injective? I suppose yes, because it is torsion free and graded divisible (i.e., divisible by homogeneous ...
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Confusion about injectivity of $\mathbb{Z}/m\mathbb{Z}$

It is well known that $\mathbb{Z}/n\mathbb{Z}$ is injective as a module over itself. And so because $\mathbb{Z}/n\mathbb{Z}$ is a PID (even if it isn't an integral domain, each ideal is generated by a ...
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Projective module on the ring of dual numbers

Let $\mathbb{K}$ be a field and consider the ring of dual numbers $R=\mathbb{K}[x]/<x^2>$. I have to prove that any projective $R$-module $P$ is injective. My idea is to use the Baer’s criterion....
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Constructing injective resolution on big sites.

Let's work over a category of varieties on some field. Let $\mathcal{F}$ be a presheaf on varieties that satisfies Zariski/etale sheaf conditions whenever restricted to the opens of a specific variety....
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Hom and Injective envelope

let $R$ be Noetherian ring and $m$ , $n$ are maximal ideals of $R$ s.t. $m\neq n$. can I say that ‎$$Hom_R(E_R(R/m),E_R(R/n))=0?$$ why?
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chain map from a resolution to another injective resolution induced by linear map is unique up to chain homotopy equivalence

Let $0\rightarrow M\rightarrow E_0\rightarrow E_1\rightarrow...$ be a resolution of a $R-$module $M$. Let $0\rightarrow N\rightarrow I_0\rightarrow I_1\rightarrow...$ be a injective resolution of a $R-...
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What is the importance of a projective cover and injective hull for a module?

My understanding of projective covers and injective hulls for modules over a (finite-dimensional) associative $\mathbb{C}$-algebra $A$ is as follows. $\bullet$ The projective cover of $M$ is an ...
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Finite injective resolution of a constant sheaf

Is it known if the injective resolution of the constant sheaf $\mathbb C_X$ on a smooth manifold $X$ is of finite length? I am asking this because the fine resolution of $\mathbb C_X$ in terms of the ...
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Difference between the injective object in the category of cochain complexes of modules and injective cochain complexes

There is a question completely ignored in the literature which is very strange considering how natural this question arises. I regard this as an omission and I hope to be able to clarify this matter ...
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Injectivity of the ring of formal power series over itself

It is well-known that the ring of formal power series $F[[t_1, ..., t_n]]$, where $F$ is a field of characteristic zero, is an injective abelian group. Is it true however that $F[[t_1, ..., t_n]]$ is ...
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Injective module is stalkwise injective

Let $X$ be a Hausdorff locally compact topological space, $R$ be a sheaf of $\mathbb{C}_X$-algebras on $X$. If necessary, we can assume that $(X,R)$ is a complex manifold. If $M$ is an injective ...
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Proof of Theorem II, 7.11 in "Residues and duality"

When reading the proof of Theorem II, 7.11 in "Residues and Duality" by R. Hartshorne, I have encountered some steps that are unclear to me. 1.Firstly, I am unsure about the notation $\...
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Injective module iff is always a direct summand

I want to prove that an R-Module $J$ is injective if and only if whenever $J$ is a submodule of $M$, $J$ is a direct summand of $M$. One implication is easy: If $J$ is injective and is a submodule of $...
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Module homomorphisms that factor through an injective module

Let $R$ be a commutative unital ring, $M$ and $N$ be $R$-modules, and $u:M\to N$ be an $R$-linear map. Let $i_M:M\to I_M$ and $i_N:N\to I_N$ be injective hulls for $M$ and $N$ and set $\Sigma M=\text{...
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injective envelope and $Hom$

Let $R$ be a commutative Noetherian ring and $I$ be an ideal of $R$. If $p$ is a prime ideal of $R$, is the following statement true: $$Hom_R(R/I,E_R(R/p))=‎ ‎\begin{cases} E_{R/I}(R/p)~~ if \hspace{....
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example of injective module

Can you provide some examples of a commutative Noetherian local ring $(R,m,k)$ and an $R$-module $M$ that is not finitely generated, such that $id_R(M)\neq \sup \{i\in \mathbb{N}_0 \mid Ext_R^i(R/m,M)...
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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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Injective object in small module category

Consider a ring $R$ and its module category $R-\mathrm{Mod}$. In a full exact subcategory $\cal{C}$ (whose exact sequences are exact sequences in $R-\mathrm{Mod}$), it is not necessary that an ...
S.Gau at Math's user avatar
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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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Is there an injective object in the category of all free abelian group?

The following notations is inherited from Homology. Suppose we have an exact sequence $$0\rightarrow B_{n}\rightarrow Z_{n}\rightarrow H_{n}(C)\rightarrow 0$$ where $B_{n}$ is the nth boundaries, and $...
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Injective Hull of $k$ as $k[x_1,..., x_n]$-module

Given a ring $A$ and an $A$-module $M$, there is a unique (up to isomorphism) injective $A$-module $E_A(M) $containing $M$, with the property that every injective $A$-module containing $M$ also ...
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A ring $R$ is noetherian iff whenever $S_1,S_2,\ldots$ are simple right $R$-modules then $\bigoplus_{i=1}^{\infty}E(S_i)$ is injective. [closed]

It's well-known that a ring $R$ is noetherian if and only if direct sums of injective $R$-modules are again injective. How to prove the following characterization: $R$ is noetherian $\iff$ whenever $...
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Lower shriek pushforward of injectives.

Given an open subscheme $f: U\hookrightarrow X$ and an injective etale sheaf of abelian groups $\mathcal{I}$ on $U$, then is it necessarily true that $f_!(\mathcal{I})$ is also injective? If not then ...
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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 @>&...
Daniel Donnelly's user avatar
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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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5 votes
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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 ...
pink floyd's user avatar
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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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