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

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2answers
23 views

Torsion elements of a module not a subgroup

Consider $A$ a ring, $M$ an A-module, and $Tor(M)$ being the set of torsion elements of M (that is, the set of $m \in M$ for which $am=0$ for some $a \in A\backslash \{0\}$ ) Show that $Tor(M)$ need ...
0
votes
0answers
19 views

Isomorphism between modules and submodules

Let $A$ be a ring, $M_1,\ldots,M_n$ be A-modules and $N_j$ be an $A$-submodule of $M_j$ for $1 \leq j \leq n$ Prove that $$(M_1 \times M_2 \times \cdots \times M_n)/(N_1 \times N_2 \times \cdots ...
1
vote
1answer
21 views

Restriction functors between Categories O over semi-simple Lie algebras

I have the following question: Let $\mathfrak{g}:= \mathfrak{gl}(3)$ be the general linear algebra and $\mathfrak{gl}(2) \cong \mathfrak{a} \subset \mathfrak{gl}(3)$ a sub-algebra. Let ...
2
votes
1answer
31 views

Galois comodules

I tried to figure out why Galois comodules is a generalization of several Galois aspects, but I could not? I am really interested in Comodule theory, and I am very curious to know the answer for ...
0
votes
1answer
19 views

Lifting map between finitely generated modules

Suppose $A$ is a commutative ring with unit, and $M$ is a finitely-generated module with the surjection $\pi: A^n \twoheadrightarrow M$. Let $f : M \to M$ be a module homomorphism. I am trying to see ...
1
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1answer
18 views

Vector Spaces and Simple Modules

Let $G$ be a finite group and let $R = \textbf{R}[G]$ be the group ring of $G$ with coefficients in the field $\textbf{R}$ of real numbers. Let $V$ be an $R$-module which is finite-dimensional as an ...
1
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1answer
19 views

Semisimple module example

I need to find an example of a module over $\mathbb{F}[x]$ which is two dimensional over the field $\mathbb{F}$ and not semisimple. I do not know how to do it. Thanks
1
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1answer
32 views

Simple and semi-simple over $\mathbb{Z}$

What is a necessary and sufficient condition on an integer $n$ for $\mathbb{Z}/n \mathbb{Z}$ to be simple as a module over $\mathbb{Z}$? Semisimple? In the case of simple I think that because it is a ...
1
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1answer
56 views

What does a complex of modules mean?

I try to understand from Qing Liu's book Algebraic Geometry and Arithmetic Curves the problem 1.2.16. It goes as follows: Let $(A,\mathfrak m)$ be a Noetherian local ring, and $$C^\bullet:0\to ...
1
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0answers
39 views

Clarification about the definition of free module

I am reading this notes. Definition 1: Let $R$ be a commutative ring with $1$. Let $S$ be a set. A free $R$-module $M$ on generators $S$ is an $R$-module $M$ and a set map $i:S\rightarrow M$ ...
4
votes
0answers
21 views

Nilpotent Jacobson radical of $End(M)$

If $M$ is a Noetherian injective left $R$-module, is it true that the Jacobson radical $J=J(End(M))$ of the endomorphism ring of $M$ is nilpotent? I know that if a left $R$-module is Artinian or ...
5
votes
1answer
39 views

A free direct sum of a projective module

I want to prove that a left module $_RP$ is a projective generator if and only if a direct sum of (copies of) $P$ is free. My try is, first, to observe that $P$ is a generator if and only if for ...
0
votes
0answers
23 views

Semi-simplicity of tensor product of simple modules over $\mathfrak{sl}(2)$

I have the following questions: $\bf Notations$: Let $\mathfrak{h} \subset \mathfrak{sl}(2)$ be a Cartan subalgebra, and $W$, $V$ be $\mathfrak{h}$-semisimple (i.e. they are weight modules), simple ...
1
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0answers
24 views

Module product and coproduct

Completely lost when reading this: "Defintion: If the index set $T$ is finite, the coproduct and product are the same module. If $T$ is infinite, the coproduct or sum $$\coprod_{t\in ...
4
votes
2answers
65 views

How to show that $3\mathbb{Z}/15\mathbb{Z} \cong \mathbb{Z}/5\mathbb{Z}$?

How to show that $3\mathbb{Z}/15\mathbb{Z} \cong \mathbb{Z}/5\mathbb{Z}$ as $\mathbb{Z}$-module over $\mathbb{Z}$? My proof: Define surjective function $f:3\mathbb{Z}/15\mathbb{Z} \rightarrow ...
3
votes
0answers
26 views

Image of homomorphism with kernel $N$ isomophic to $M/N$

Let $M$ be an $R$-module and $N$ be a submodule. Let $f:M\rightarrow M'$ be a surjective homomorphism with kernel $N$. How to show that $M'$ is isomorphic to $M/N$? My thought: Define ...
1
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0answers
25 views

Understanding Tensor product of modules

This is follow up question Understanding the Details of the Construction of the Tensor Product If $M$ and $N$ are 2 A-modules, we define $Z = A^{M\times N}$ as module generated by $M\times N$. Then ...
1
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1answer
25 views

To show module homomorphism being injective

Suppose $R$ is a field, $M$ is a right $R$ module, and $f : R_R \rightarrow M$ is a non-zero homomorphism. Show that $f$ is injective. My work: The homomorphism is uniquely determined by $f(1)$. If ...
3
votes
0answers
39 views

Flatness of integral closure over an integral domain

The problem 1.2.10 from Qing Liu's book "Algebraic Geometry and Arithmetic Curves" is the following: Let $A$ be an integral domain, and $B$ its integral closure in the field of fractions ...
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0answers
30 views

Is my observation correct regarding restriction of scalars?

Let $\alpha: \Lambda\to \Gamma$ be a ring homomorphism, then $ _\Lambda\Gamma_\Gamma$ is a bimodule. We have the following pairs of adjoint functors $$ \mathbf{Mod_\Lambda} \xrightarrow{\cdot\; ...
0
votes
1answer
37 views

Tensoring the exact sequence by a faithfully flat module

I have problems to do the exercise 1.2.13 from Liu's "Algebraic Geometry and Arithmetic curves". First, Liu defines that if $M$ is a flat module over a ring $A$, then $M$ is faithfully flat over $A$ ...
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votes
0answers
27 views

Multiple products of submodules

NOTE: This is part of a homework, so only worry about the question regarding notation. We have the following conditions: $R=\mathbb{Z}$, $I = \mathbb{Z}_{>0}$, and $M_i = \mathbb{Z} / i ...
2
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1answer
29 views

The dual of a tensor algebra as a right module

Let $V$ be a finite-dimensional vectorspace over a field $k$, and let $T$ denote the tensor algebra of $V$ (thought as a graded $k$-algebra). Denote by $T^\vee$ the dual of $T$, i.e. ...
2
votes
1answer
30 views

$V^{\oplus3}$, linear constraints. [on hold]

Let $V$ be an irreducible $G$-representation over $\mathbb{C}$, and let $W = V \oplus V \oplus V$. Prove that all submodules of $W$ are given by "imposing linear constraints," e.g.$$\{(x, y, z) \in V ...
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votes
0answers
33 views

$V$ is $G$-irrep. over $\mathbb{C}$, submodules of $V \oplus V \oplus V$ given by imposing linear constraints. [closed]

Let $V$ be an irreducible $G$-representation over $\mathbb{C}$. Let $W = V \oplus V \oplus V$. Show that all submodules of $W$ are given by "imposing linear constraints," e.g.$$\{(x, y, z) \in V ...
5
votes
1answer
29 views

Is annihilator of maximal submodule is a maximal ideal?

Let $R$ be a commutative ring with identity element and $M$ is an $R$-module. We know the annihilator of a submodule $N$ of $M$ ($I=(N:M)$) is an ideal in $R$. If $N$ is a maximal submodule of $M$, is ...
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0answers
38 views

Is Hom(F, I) an FP-injective module? [on hold]

This question can be in the advanced abstract algebra, I think this is right. In fact, I have deleted the tage-abstract algebra. I am confused for the "off-topic", I hope it is right now. Let f be a ...
1
vote
1answer
29 views

Flatness of module over field of fractions

This is from Liu 1.2.9. Let $A$ be an integral domain, and $K$ its field of fractions. Let $M$ be a finitely generated sub-$A$-module of $K$. Why do $M$ is flat if and only if $M_{\mathfrak p}$ is ...
3
votes
2answers
52 views

Suppose $M$ is a Noetherian $R$-module and $\phi: M \rightarrow M$ is an $R$-module endomorphism of $M$.

So I did an exercise in my algebra textbook which was to show that $\ker(\phi^n) \cap \operatorname{im}(\phi^n) = 0$ and show that if $\phi$ is surjective, then $\phi$ is an isomorphism. I thought to ...
1
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1answer
23 views

Characterizing the Prüfer $p$-group

I've been trying to solve these questions for the past few hours with no luck: If $G$ is an infinite abelian group all of whose proper subgroups are finite, then $G$ is a Prüfer $p$-group for some ...
1
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4answers
78 views

An elementary fact about tensor product of modules

Let $A$ be an $R$-right module, $N$ be a submodule of $R$-left module $M$, $\pi:M\rightarrow M/N $ is the natural epimorphism. What is $\ker\left(\pi\otimes1_{A}\right) $? I appreciate your help.
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0answers
24 views

A PID is an indecomposable as a module over itself [closed]

Let $R$ be a PID. And $R = I + J $, where $I,J \neq 0$ are ideals. How do I show that $I \cap J \neq 0$?
1
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1answer
32 views

Is ring R itself a finitely generated module over $R$?

It seems trivial that ring $R$ itself is a $R$-module. But then can we say R is finitely generated by multiplicative identity? That seems so trivial..
0
votes
1answer
49 views

Canonical homomorphism and free module, Liu 1.2.8 c

How can I do the problem 1.2.8 c in "Algebraic Geometry and Aritmetic Curves". Namely, let $A$ be a Noetherian ring, $M$ a finitely generated $A$-module, and $N$ an $A$-module. Let $B$ be a flat ...
4
votes
1answer
77 views

Classification of indecomposable modules over a given ring

Let $K$ be a field, how can I classify all the indecomposable modules up to isomorphism over the ring $R = K [x] / (x^n ) $?
2
votes
1answer
56 views

How to prove that a ring is not flat over $k[t,s]$? [duplicate]

Let $k$ be a field, $A=k[t,s]$, and $C=A[z]/(tz-s)$. How can I prove, using the ideals $tA$ and $sA$, that $C$ is not flat over $A$? I know that if $A$ is a Dedekind domain then $A$-module is ...
0
votes
1answer
42 views

$H^i_I(M)$ is finitely generated iff the support of $Ext^{d-i}_S(M, S)$ has dimension zero

$(R,m)$ is a local Noetherian ring. $M$ is a finite $R$-module. Here, using dualizing complex, Karl Schwede says that if $R=S/I$ where $S$ is regular of dimension $d$, then we have: "$H^i_m(M)$ is ...
2
votes
1answer
52 views

An example of $_AM$ is simple but $M^\star_A$ is not?

Let $A$ be a finite dimensional $k$ algebra, $k$ is a field. It is evident that the duality functor ($(-)^\star=hom_k(-,k)$) preserves the simplicity in the case of finitely generated $A$- modules ...
1
vote
1answer
49 views

Question about split monomorphisms of free modules over local rings

In May's notes on Cohen-Macaulay and Regular Local Rings, during the proof of Serre's theorem on page 9, he claims that if $R$ is a local ring and $\phi\colon F\to F'$ is a map of finitely ...
1
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1answer
24 views

Canonical homomorphism related to ideal is an isomorphism

I have a problem to do the exercise 1.2.1 b on Liu. Namely, Let $M$ be an $A$-module, $I\subseteq \operatorname{Ann}(M)$ an ideal, $N\ne M$ is an $A$-module such that $I\subseteq ...
0
votes
0answers
32 views

Classifying modules over $\mathbb{Z}_2[x]$ with precisely 8 elements

I would just like to verify I have the right idea here. The problem is to classify up to isomorphism all modules having precisely 8 elements over $\mathbb{Z}_2[x]$. Since $\mathbb{Z}_2[x]$ is a PID, ...
2
votes
3answers
63 views

Exact sequence with flat module tensored by module stays exact

The following theorem is given in Liu proposition 1.2.6: Let $A$ be a ring. Let $0\to M^\prime\to M\to M^{\prime\prime}\to 0$ be an exact sequence of $A$-modules. Let us suppose that ...
2
votes
3answers
26 views

Ideal contained in annihilator problem

Let $M$ be an $A$-module, and $I\subseteq \operatorname{Ann}(M)$ be an ideal. Why do we can endow $M$ in a natural way with the structure of an $A/I$-module, and why do $M\simeq M\otimes_AA/I$?
4
votes
1answer
45 views

Existence of a non-semisimple ring such that every module over it has a simple submodule [closed]

Does there exists a ring $R$ which is not semisimple but every module over it has a simple submodule?
1
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0answers
98 views

Prove that factor modules are isomorphic.

I'm trying to prove (from a previous post) that if $A=k[x,y,z]$ and $I=(x,y)(x,z)$ then $((x,y)/I)/((x,yz)/I) \cong A/(x,z)$. I did this by defining the homomorphism $\phi: A \to ...
2
votes
1answer
29 views

Existence of a canonical surjective homomorphism on tensor products

I got stuck with Liu's "Algebraic Geometry and Aritmetic Curves" exercise 1.1.7. Let $B$ be an $A$-algebra, and let $M$, $N$ be $B$-modules. Why do there exists a canonical surjective homomorphism ...
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0answers
36 views

Free module as a direct sum

Let $ F$ be a free $R$-module,with basis $w_1,..,w_n$. Then for $x \in F $ we have that $ x=w_1r_1+...+w_nr_n$ with $r_i \in R$ ,and $w_1,..,w_n$ are linearly independent over $R$. I have to prove ...
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vote
0answers
46 views

Liu 1.1.5, extension of scalars

I tried to do the exercise 1.1.5 on Liu's Algebraic Geometry and Arithmetic Curves. Namely, Let $\rho:A\to B$ be a ring homomorphism, $M$ an $A$-module, and $N$ a $B$-module. How can I show that ...
-1
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0answers
31 views

Meet of two submodules may not be a submodule.

Can someone give me an example in support of the following fact? The meet of two submodules may not be a submodule.
1
vote
1answer
41 views

Artinian module and short exact sequence

Consider the short exact sequence of modules $$0\rightarrow M\rightarrow N\rightarrow K\rightarrow0$$ Let $K$ be an Artinian (Noetherian) module, can we get $N$ is an Artinian (Noetherian) module?