# 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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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 @>&... • 19.3k 1 vote 1 answer 49 views ### 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:... • 2,901 1 vote 1 answer 36 views ### 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 ... • 599 1 vote 1 answer 20 views ### 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 ... • 3,779 4 votes 0 answers 88 views ### 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 ... • 1,156 0 votes 1 answer 29 views ### 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 ... 0 votes 0 answers 19 views ### 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 ... • 1,194 2 votes 2 answers 80 views ### 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,... • 2,901 0 votes 0 answers 22 views ### 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 \... • 599 9 votes 0 answers 186 views ### 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 ... • 91 0 votes 1 answer 107 views ### 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 ... • 599 1 vote 0 answers 57 views ### 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 ... 4 votes 1 answer 70 views ### 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 ... • 2,290 1 vote 0 answers 23 views ### 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 ... • 412 3 votes 1 answer 81 views ### 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)... 0 votes 0 answers 18 views ### 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)... 3 votes 1 answer 225 views ### \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 ... • 10.8k 2 votes 1 answer 36 views ### 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-... 0 votes 0 answers 21 views ### 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 ... • 1,096 1 vote 0 answers 40 views ### 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 ... • 68 1 vote 1 answer 71 views ### 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 ... 0 votes 0 answers 35 views ### 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 ... • 599 1 vote 1 answer 62 views ### 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 ... • 3,006 1 vote 1 answer 34 views ### 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 ... • 175 0 votes 0 answers 42 views ### 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 ... • 243 1 vote 0 answers 34 views ### 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 ... • 243 7 votes 3 answers 404 views ### 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 ... • 716 0 votes 0 answers 51 views ### 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 ... • 21 0 votes 1 answer 52 views ### 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 ... 1 vote 1 answer 39 views ### 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 ... • 23 0 votes 0 answers 116 views ### 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? 0 votes 0 answers 58 views ### 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 ... • 55 0 votes 1 answer 72 views ### 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]/ \... • 485 0 votes 1 answer 103 views ### 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. ... • 485 1 vote 0 answers 41 views ### 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 ... • 21 0 votes 0 answers 25 views ### 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 ... • 1,242 0 votes 1 answer 37 views ### 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: (\... • 21 1 vote 1 answer 28 views ### 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 ... • 599 3 votes 1 answer 106 views ### 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&... • 100 5 votes 0 answers 97 views ### 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}... • 8,472 1 vote 1 answer 32 views ### 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 ... • 502 2 votes 0 answers 134 views ### 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 ... • 4,614 0 votes 0 answers 33 views ### 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 ... • 876 2 votes 0 answers 62 views ### 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,... • 876 1 vote 2 answers 93 views ### 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-... • 876 2 votes 2 answers 115 views ### 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 ... • 1,065 2 votes 1 answer 95 views ### \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 ... 44 views

### 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 ...