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Questions tagged [modules]

For questions about modules over rings, concerning either their properties in general or regarding specific cases.

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Graded endomorphisms factoring through nilpotent ones

Let $M$ be a finitely generated 2-graded $k[x,y]$-module, concentrated in nonnegative degrees, and $f$ be a degree $(n,m)$-endomorphism of $M$. We can consider $f$ as a morphism $M\langle n,m\rangle\...
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what is the structure of $R \otimes \mathbb{Q}[[X]]$?

Let $R$ be a ring and $\mathbb{Q}[[X]]$ be the ring of formal power series in rational field $\mathbb{Q}$. Let $f(X) \in R \otimes \mathbb{Q}[[X]]$ be a power series in $X$. My question is- How does ...
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1answer
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Cancelling finitely generated projective modules from a tensor product of finitely generated projective modules

Let $R$ be a commutative Noetherian ring (with unity) and $M,N,P$ be finitely generated projective modules over $R$ such that for some $n\ge 1$, we have $M\otimes_R N \cong M \otimes_R P \cong R^n$. ...
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Proving that the endomorphism ring of a projective module is semiperfect

Let $_RP$ be a projective module, then the following are equivalent: Every quotient of End($_RP$) has a projective cover (i.e it is semiperfect) Every quotient of P has a projective cover (i.e P is ...
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1answer
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Is $\{ 0 \}$ a basis of the free module $\{ 0 \}$?

I'm studying modules by reading Dummit and Foote, and I'm having a problem understanding the definition of a free module. I read this stackexchange question, but I couldn't figure it out. The ...
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How can one express $\mathbb{Q} / \mathbb{Z}$ as a direct sum of $\mathbb{Q}$ and $\mathbb{Z} / p$ for $p$ prime?

Consider the $\mathbb{Z}$ modules $\mathbb{Q}$ and $\mathbb{Z} / p$ for $p$ prime. I have a result that says that every injective $\mathbb{Z}$ module is a direct sum of these modules. I also know that ...
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1answer
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Isomorphism between tensor product of modules and quotient module

I'm trying to show that $M\otimes_{R}R/\mathbf{m}$ and $M/\mathbf{m}M$ are isomorphic as $R$-modules, where $M$ is an arbitrary $R$-Module, $R=k[x_{1},\ldots,x_{n}]$ with a field $k$, $\mathbf{m}=(x_{...
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1answer
25 views

Proving basic properties of $\mathbb{C}G$-modules

Is my following proof correct? Let $G$ be a finite group and $\mathbb{C}G$ its group algebra. Let $\phi:\mathbb{C}G\to\mathbb{C}G$ be a $\mathbb{C}G$-homomorphism. $ i)$ Then there exists a $w\in\...
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1answer
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Is a finitely generated module over the field of fractions is also finitely generated over the original integral domain?

Let $R$ be an integral domain and $F$ its field of fractions. Let $M$ be a finitely generated $F$-module. Question: Is $M$ also a finitely generated $R$-module? I know that $M$ is an $R$-module ...
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Question about associated primes and annihilator

I am trying to solve the following exercise: Let $R$ be noetherian and $M$ a finitely genererated $R$-module. Show that $\mathrm{Ass}(R/\mathrm{Ann}(M)) \subseteq \mathrm{Ass}(M)$ and both sets ...
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On covariant linear functors $T: R$-Mod $\to R$-Mod which preserves direct-limits or inverse-limits

Let $R$ be a commutative ring with unity. Let $R$-Mod denote the category of $R$-modules, and $Ab$ denote the category of Abelian groups. Now, it is known that a covariant additive functor $T: R$-...
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1answer
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Does every covariant, additive, faithfully-exact functor $T:R$-Mod $\to Ab$ preserve either direct sum or direct product?

Let $R$ be a commutative Noetherian ring. Let $Ab$ denote the category of abelian groups. Let $T:R$-Mod $\to Ab$ be a covariant, additive functor such that for any sequence of $R$-modules, $A \...
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On faithfully flat and faithfully projective modules

Let $R$ be a commutative Noetherian ring. Let $P,Q$ be some $R$-modules such that $-\otimes_R P $ and $ Hom_R(Q,-) $ are faithfully exact functors i.e., for any sequence of modules $A \xrightarrow{f}...
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1answer
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Does taking the radical of modules commute with taking quotients?

I am studying a proof which shows that a particular $R$-module map $\pi$ is surjective onto a module $M$. The details of the map are complex, so I won't give them here, and will just sketch what I ...
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$(\mathfrak{g},K)$-Module where $K$ is a field?

I am reading through Finite groups of Lie type_ conjugacy classes and complex characters by Roger Carter, and came across this passage where Carter is setting up a special class of module to give a ...
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The class of left serial rings is closed under extensions

A class $S$ is closed under extension if given an ideal $I \subseteq R$ such that $I\in S$ and $R/I\in S$, then $R\in S$. A ring $R$ is left serial if it is a direct sum of left uniserial rings. ...
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Is a finitely generated torsion-free R-module free over R if R is an integral domain?

I know this is the case if $R$ is a PID, but PID's are special instances of Integral Domains, so I am wondering if there is a counter-example to the case where R is an integral domain. This post ...
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1answer
44 views

Why is $V^{\vee}\otimes W^{\vee}\longrightarrow (V\otimes W)^{\vee}$ always injective?

Let $R$ be a commutative ring with $1$. For all $R$-modules $V,W$ we have a canonical $R$-linear map $V^{\vee}\otimes W^{\vee}\longrightarrow (V\otimes W)^{\vee}$ from tensor product of dual modules ...
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Local behavior of sheaf of ideals given by a closed immersion

I know that if $Y \hookrightarrow X$ is a closed embedding i of schemes, then the sheaf of ideals $I_Y(U) = $ {$f \in \mathcal{O}_X(U)\text{ } | i^*(f) = 0$} is quasi coherent. I sort of understand ...
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Given $\phi \in \mathrm{End}(M)$, when does $\phi$ injective imply $\phi^*$ surjective and $\phi^*$ injective imply $\phi$ surjective?

Let $M$ be a module over a commutative ring (with unity) $R$. Let $\phi : M \to M$ be an $R$-module homomorphism. Then we have a dual map $\phi^* : M^* \to M^*$ given by $\phi^*(f)=f\circ \phi, \...
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Bijective $\mathcal{O}_X$-Module Homomorphisms

Let $(X,\mathcal{O}_X)$ be a ringed space, $\mathcal{F}$ and $\mathcal{G}$$\,\,\,$$\mathcal{O}_X$-modules, and $\varphi:\mathcal{F}\to\mathcal{G}$$\,$ an $\mathcal{O}_X$-module homomorphism. If $\...
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Given a split exact sequence $0 \to N \to M \to M \to 0$, when can we say $N=0$?

Let $M$ be a module over a commutative ring $R$. Let $N$ be a submodule of $M$ such that there is a split exact sequence $0 \to N \to M \to M \to 0$. So, in particular, $M \cong M \oplus N$. Under ...
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1answer
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Finitely generated projective resolution of a module over a regular local ring

Let $R$ be a regular local ring and let $M$ be an $R$-module. Then there exists a finite projective resolution $P_\bullet\to M\to 0$. However, need there exist a finite projective resolution ...
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1answer
35 views

Hopfian modules and equivalence of categories of modules

For a ring with unity (not necessarily commutative) $R$, let $R$-$Mod$ denote the category of left $R$-modules. Let $R,S$ be two rings with unity and $T: R$-Mod $\to S$-Mod be an equivalence of ...
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How does Matlis duality behave w.r.t. Hopfian and Co-hopfian modules?

Let $(R,\mathfrak m, k)$ be a Noetherian, complete, local ring. Let $E$ be an injective hull of $k$. We know that the Matlis duality functor $D(-):= Hom_R(-, E)$ gives an anti-equivalence between the ...
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Torsion-less module over commutative ring whose injective hull is Hopfian

Let $M$ be a module over a commutative ring (with unity) $R$. Let $E_R(M)$ denote the injective Hull of $M$ . If $M$ is torsion-less (i.e. $\cap_{f\in M^*=Hom_R(M,R)} \ker f=(0)$ ) and $E_R(M)$ is ...
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Two definitions of torsion pairs/theories. How are they equivalent?

Let $\mathcal{A}$ be an abelian category, $\mathcal{T}, \mathcal{F}$ two strictly full additive subcategories of $\mathcal{A}$. Then according to nLab and other sources including Constructing Torsion ...
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A trivialization of line bundle is same as nonvanishing section

I am reading Lemma 17.22.10. My fundamental confusion is how is Nakayama Lemma applied. Claim: Let $X$ be a ringed space. Assume that each $O_{X,x}$ is a local ring with maximal ideal $m_x$. Let $ ...
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Endomorphism rings of indecomposable modules

This is a more structured reformulation of this question. Let $k$ be a field and $A$ be a commutative $k$-algebra, say $A=k[x_1,\dotsc,x_n]$, and $M$ be a finite dimensional (ungraded or $n$-graded) $...
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On Hopfian modules over commutative Noetherian rings

Let $R$ be a commutative Noetherian ring with unity. Let us call an $R$-module $M$ to be Hopfian if every surjective endomorphism $M \to M $ is injective. 1) If $M_1$ and $M_2$ are Hopfian modules, ...
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Long exact sequence of cohomology

I was reading about cohomology and long exact sequences. I found that Given $$0 \to L \to M \to N \to 0$$ is a short exact sequence of $G$- modules, then a there exists a long exact sequence is ...
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Is the Artinian property dual to the Noetherian property?

If $R$ is a ring and $M$ is a left $R$-module, we say that $M$ is Noetherian whenever it satisfies any of the equivalent conditions: 1N. Every ascending (under subspace inclusion) chain of submodules ...
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Left Exactness of global sections functor over quasi compact sheaves [duplicate]

Let $0 \rightarrow E' \rightarrow E \rightarrow E''$ be a short exact sequence of quasi coherent sheaves on a scheme X. Show that the sequence $0 \rightarrow E'(X) \rightarrow E(X) \rightarrow E''(X)$ ...
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1answer
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Proof about length of quotient modules

Let $M$ be an $R$-module and let $x, y \in R$ such that $y$ is not a zero divisor of $M$ and $M / xyM$ has finite length. Show that $l(M/xyM)=l(M/xM)+l(M/yM)$. In the above, $l$ denotes the lenght ...
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1answer
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Is the Evaluation map of an R-Module of rank 1 and hom injective

This is in context of a larger problem of showing that the dual of an invertible sheaf is invertible on a scheme. I want to show that given a free R-module A of rank 1, the standard evaluation map ...
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Is the ring $R$ a topological ring with respect to the following topology?

Background This question is motivated by trying to answer this question. But before going into the question straight let me give some background. Definition 1. Let $R$ be a ring and $A$ be an ...
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2answers
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how to prove that $\mathbb Q$ is flat as a $\mathbb Z-$module [duplicate]

I know that $Tor^{\mathbb z}_1(\mathbb Z, N) = 0$ for any $\mathbb Z-$module, because free modules are flat. Then because $Tor_1$ is local, we have $Tor_1^{\mathbb Q}(\mathbb Q, S^{-1}N) = 0$, which ...
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1answer
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Let $R$ be a local ring with a nilpotent maximal ideal $M$ and $I\subseteq M$. Then $ya\in I$ implies $ya=0$ for some $0\neq y\in R$

Let $R$ be a local ring with a nilpotent maximal ideal $M$. If $I\subseteq M$ is a fixed ideal for which a fixed element $a\in R$, $a+I\neq I$. Prove that for any $0\neq y\in R$ and $y\notin I$, $...
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1answer
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Indecomposable module with $\operatorname{End}(M)$ non-commutative

Let $A$ be commutative $k$-algebra over a field $k$, and $M$ be a finite dimensional indecomposable* $A$-module. In general, $\operatorname{End}_A(M)/J(\operatorname{End}_A(M))$ will be a division ...
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1answer
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What is the $\oplus$ sign mean in $R^{\oplus I}$?

I read a defenition of a "finite generated module": An $R$-module $M$ is said to be finite generated if there exists a surjective homomorphism $R^{\oplus I}\to M$ for some finite set $I$. For $I=\{1,....
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1answer
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The exact sequence of tensor product

Prove that for all free right $R$-module $F$ and for all exact sequences of $R$-modules $$0\to M\xrightarrow{f}N\xrightarrow{g}P\to 0$$ then $$0\to F\otimes_RM\xrightarrow{1\otimes f}F\otimes_RN\...
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A problem about checking isomorphism of R-module

Let $R=K\left[x_1,x_2,\cdots,x_n\right]$ be a polynomial ring with coefficients in the field $K$; $\alpha_1,\alpha_2,\cdots,\alpha_p,\beta_1,\beta_2,\cdots,\beta_q\in R^{1\times m}$; $M_1$ be the $R$-...
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1answer
25 views

Example of both finitely and infinitely generated free modules which are direct sum of two non free modules

Does there exist non free $R$-modules $F_0,F_1$ such that $F=F_0\oplus F_1$ be a free $R$-module? 1- If yes then for what kind of rings $R$ there exist such $R$ modules? 2- If yes then does it holds ...
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Determinant of a nonfree module

Is there a definition of a determinant which can be applied to a module with no basis? We can produce a module with noncommutative rings, without knowing a basis for these rings, i.e. without units. ...
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Condition for a finitely generated flat module be projective

Prove that: Let $R$ be a commuatative ring, let $T$ be total quotient ring of $R$. A finitely generated flat $R$-module $M$ is projective if and only if the scalar extension $T\otimes_R M$ is a ...
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1answer
29 views

Why is $M/\operatorname{rad}(M)$ semisimple?

Let $M$ be an $A$-module for $A$ a finite dimensional algebra. Let $\operatorname{rad}(M)=\bigcap\{N\subsetneq M\ \text{maximal}\}$. Clearly, $M/N$ is simple for any maximal submodule $N$. It seems to ...
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1answer
55 views

Isomorphic algebras of endormporphisms

Let $R$ be a simple algebra, $M$ a simple $R$-module and $N$ a simple $\mathcal{M_n}(R)$-module (always considering finite dimension). Prove that $End_R(M)$ and $End_{\mathcal{M_n}(R)}(N)$ are ...
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1answer
38 views

Exact sequence construction

Given an $R$-module $M$ arbitrary, show it is always possible to construct an exact sequence of $R$-modules $$0\longrightarrow K \longrightarrow L \longrightarrow M \longrightarrow 0,$$ with $L$ a ...
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1answer
23 views

Cardinality of an intersection of two submodules.

Assume $p$ is a prime number and $q = p^2$. Denote by $A$ the ring $\mathbb{Z} / q \mathbb{Z}$. Consider a finite type module $M$ over $A$ whith cardinality $q^N$ where $N$ is an integer, $N>0$. ...
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28 views

Proof that $\Omega_{S^{-1}B/A}=S^{-1}\Omega_{B/A}$

The notations are: $\varphi:A\to B$ is a ring map, $S\subset B$ is a multiplicative subset of $B$, and $\Omega_{B/A}$ is the module of Kähler $A$-differentials of $B$. In a proof of the fact that $\...