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

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2
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0answers
10 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 ...
0
votes
0answers
13 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 ...
0
votes
0answers
17 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 ...
3
votes
2answers
60 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 ...
2
votes
0answers
22 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
vote
0answers
23 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 ...
0
votes
1answer
21 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 ...
2
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0answers
33 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 ...
1
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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
votes
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
27 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 ...
-2
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0answers
33 views

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

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 ...
1
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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
28 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
51 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
vote
1answer
22 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 ...
2
votes
4answers
64 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.
-4
votes
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
vote
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
43 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
75 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
45 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
vote
1answer
23 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
62 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
24 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
44 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
vote
0answers
95 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
28 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 ...
1
vote
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
votes
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?
0
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0answers
29 views

Elementary Divisors on a PID

Let $N$ be a submodule of $\mathbb{Z}^3$ generated by $\{e_1-e_3,2e_1+3e_2+e_3,3e_1+e2+5e_3\}$, with $\{e_1,e_2,e_3\}$ the canonical basis. I am asked to compute a base for $N$ by the structure ...
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0answers
25 views

Without loss of generality $P_1, . . . , P_s$ contain all elements of $gr_I (R)$ of positive degree, and $P_{s+1}, . . . , P_r$ do not

In the $\underline {Proposition\ 8.5.7}$ of book: Integral Closure of Ideals, Rings, and Modules, (Irena Swanson and Craig Huneke), they say: "Without loss of generality $P_1, . . . , P_s$ contain all ...
2
votes
2answers
45 views

Tensor product of two finitely generated modules

How can I show that if $M$ and $N$ are finitely generated $A$-modules, then so is $M\otimes_AN$? I understand that I have assumption that there are integers $n,m$ such that there are surjections ...
1
vote
2answers
50 views

Can we take a tensor product of algebra and module?

I'm trying to learn tensor product and I found that there are at least two different tensor products, tensor product of modules and tensor product of algebras. But can we mix them? Like if $M$ is an ...
3
votes
1answer
46 views

A little exercise about free modules

Let $R$ be a non-zero ring with unity. Let $a$ be a proper ideal of $R$. I wish to prove the following statement: $R/a$ is a free left module over $R$ iff $a=0$ where $R/a$ has the usual scalar ...
3
votes
1answer
35 views

Finite generation of modules

Let $M$ be an $R$-Module. Suppose we know that $M$ is finitely generated. Let $X\subseteq M$ be any generating set. Is there a finite subset of $X$ that generates $M$? I stumbled about this when ...
0
votes
1answer
58 views

On a complex group rings of finite groups

Let $G$, $K$ and $H$ be three finite groups where $H$ acts on both of $G$ and $K$ as sets nontrivially. Moreover assume that $\mathbb{C}K$ is (isomorphic to) a $\mathbb{C}H$-submodule of $\mathbb{C}G$ ...
0
votes
0answers
45 views

If M is a free R module then $M \otimes S$ is a free S module.

Question is to prove that : If M is a free R module then $M \otimes S$ is a free S module, where S is an R-algebra via $\phi$. What i have done so far is : If M is a free and finitely generated R ...
0
votes
1answer
44 views

What is $\mbox{Hom}(\mathbb{Z},\mathbb{Z}/n\mathbb{Z})$?

What is $\mbox{Hom}(\mathbb{Z},\mathbb{Z}/n\mathbb{Z})$? Will it be cyclic always? I cannot find it explicitly but I understand that there are $d(n)(=$ Number of divisors of $n)$ number of elements.I ...
1
vote
0answers
24 views

Does the tensor product of a finitely presented module and a flat module always finitely presented?

If M is an R-module which admits a degreewise finite projective resolution (i.e., a projective resolution P of M such that each Pi in P is finitely generated projective) and N a flat R-module, does ...
2
votes
2answers
54 views

A rank-nullity theorem between $\mathbb Z^n$ and $\mathbb Z^k$ [duplicate]

I think this is correct: If $\phi:\mathbb Z^{n}\to\mathbb Z^{k}$ is a group homomorphism then $n=\operatorname{rank}\operatorname{im}\phi+\operatorname{rank}\ker\phi$. Here is my attempt at a ...