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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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1answer
30 views

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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42 views

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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14 views

Prove that the following conditions are equivalent for a $_RP$ projective module

Let $_RP$ be a projective module, then: End($_RP$) is semiperfect P is semiperfect and finitely generated are quivalent. I have to prove this, but I think I'm not understanding the idea behind. ...
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1answer
37 views

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
25 views

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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24 views

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
10 views

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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12 views

$(\mathfrak{g},K)$-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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20 views

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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1answer
16 views

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
38 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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23 views

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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36 views

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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1answer
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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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0answers
64 views

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
13 views

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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1answer
30 views

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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1answer
42 views

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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0answers
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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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1answer
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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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0answers
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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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1answer
46 views

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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0answers
32 views

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
26 views

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
60 views

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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1answer
72 views

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
50 views

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
34 views

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
41 views

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 ring ...
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1answer
26 views

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
36 views

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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0answers
24 views

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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25 views

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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1answer
18 views

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
27 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 $\...
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1answer
43 views

Invertible $R$-module, $R$ local ring, $L \cong R$

In the last 5 lines of Lemma 17.22.4, the author seems to claim: If $L$ is an invertible $R$-module, and $R$ is a local ring, then $L \cong R$. The algebra section doesn't address this case. How ...
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1answer
26 views

Show that this mapping between localized modules is an isomorphism

Let $R$ be a ring. Let $M$ and $N$ be $R$-modules where $M$ is finitely presented. Then for every multliplicative set $S \subset R$ the canonical mapping $~~~~~~~~~~~~~~~~~~~~~~~~~~~~Hom_R(M,N) \...
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0answers
26 views

Showing that M is a finitely generated R-module

Lemma: Let $A$ be an $R$-Algebra such that $A$ is a finitely generated $R$-Module. Let $M$ be a finitely generated $A$-module. Show that $M$ is a finitely generated $R$-Module. Proof: So things I ...
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1answer
20 views

Ideals of Dedekind rings are projective

Let $R$ be a Dedekind domain and $I$ be an ideal of $R$. Show that $I$ is a projective $R$-module. My definition of a projective module is that it is a direct summand of a free module, i.e. there ...
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2answers
50 views

Are the two scalar multiplication on an $R$ module equivalent?

Let $R$ be a commutative ring with unity. Let $M$ be an $R$-module. Then, consider $Hom_{\mathbb{Z}}(R, M)$ (note that the homomorphisms are as $\mathbb{Z}$ modules). This is an $R$ module in two ...
2
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2answers
45 views

Isomorphic $\mathbb{C}[X]$-modules [duplicate]

My question is : It is true that $\mathbb{C}[X]/(x-c)$ and $\mathbb{C}[X]/(x-d)$ are isomorphic $\mathbb{C}[X]$-modules if and only if $c=d$? I have the feeling that the answer is simple but I ...
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1answer
26 views

If $I\subseteq R$ is a uniserial ideal of $R$ with finite length and $R/I$ is uniserial of finite length, then $R$ is uniserial of finite length

A module is called uniserial if the lattice of its submodules is a chain, i.e., the set of all its submodules is linearly ordered by inclusion. A ring is called right (resp. left) uniserial if it is ...