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

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2
votes
1answer
23 views

Why quotient ring of intersection of annihilators for a Jacobson ring is Jacobson?

Let $R$ be a commutative ring with identity and $M_i$ is a finitely generated $R$-module, for $i=1,\dots,n$. If for every $i$, $R/\operatorname{Ann}(M_i)$ is a Jacobson ring, why $R/\bigcap_{i=1}^n ...
2
votes
1answer
29 views

A question about $rad(M)$

We know that for any $R$-module $M$ the radical $rad(M)$ is the sum of all small submodules of $M$. My question: "if $x\in rad(M)$ is it true in general that $Rx$ is a small submodule of $M$? ...
2
votes
1answer
13 views

Freely generated modules and bases

I'm trying to show that a subset $S=\{m_1,\dots,m_k\}$ of an $R$-module $M$ generates $M$ freely if and only if $S$ is a basis for $M$. I think I can see the 'if' - using the unique expression of a ...
0
votes
0answers
27 views

Superfluous radical

If $M$ is a projective Artinian (left) module over a ring $R$, could one say that the radical of $M$ is a superfluous submodule in M? I know that for a projective module $M$ the radical of $M$ equals ...
6
votes
2answers
51 views

Is simple module over commutative ring always a field?

M is a simple module if and only if $M\cong R/I$ for some I maximal ideal in R. If $R$ is commutative, can I say M is a field? I'm confused about this fact because when proving it I use the fact that ...
3
votes
1answer
26 views

$Rx$ is a simple module

Let $M\neq 0$ be an $R-$module. If $M=Rx$ for each $0\neq x\in M$, why is M a simple module?
0
votes
1answer
15 views

Faitfully flatness over $B$ and flatness over $A$ equivalence

Let $B$ be an $A$-algebra, and let $E$ be a faithfully flat $B$-module. How can I show that $E$ is flat over $A$ if and only if $B$ is flat over $A$? (Liu, Algebraic Geometry and Arithmetic Curves, ...
1
vote
0answers
10 views

Finding Modules with Given Length of Composition Series

I study a course about commutative algebra, and I saw many questions as the following: Find an example of a $\mathbb{Q}\left[\lambda_1,\lambda_2\right]$-module with $\ell\left(M\right)=3$ (i.e., ...
0
votes
1answer
24 views

Injective acton of an Chevalley generator of $\mathfrak{sl}(2)$ on non-intergral weight module

I have a problem as follows. Let $E_{i,j}\in M_2(\mathbb{C})$ be the elementary matrix and $\mathfrak{g}:=\mathfrak{sl}(2) = \{ A \in M_2({\mathbb{C}})| tr(A) =0 \}$ the special linear Lie algebra. ...
0
votes
1answer
17 views

Equivalence of finitely generated of faithfully flat and annihilator

How one can show that a finitely generated flat $A$-module $M$ is faithfully flat if and only if $\operatorname{Ann}(M)=0$? (Liu, Algebraic Geometry and Arithmetic Curves, Exercise 2.17.) I tried ...
0
votes
1answer
18 views

Finding a module for the series $2^{i}$ from 0 to 219

How can I compute this: $\{ \sum 2^{i}$ for $i \in [0, 219] \} \pmod{13}$ I tried to manipulate the series by using the root principle to find the number of elements divisible by every prime $\leq ...
-1
votes
2answers
33 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 ...
1
vote
0answers
23 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
28 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
46 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
vote
1answer
20 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
vote
1answer
20 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
vote
1answer
33 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
vote
1answer
57 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 ...
3
votes
1answer
79 views
+100

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
22 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
40 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
vote
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
67 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
vote
0answers
26 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
vote
1answer
26 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
40 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
vote
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
39 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 $A$ ...
-2
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
31 views

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

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
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 ...
6
votes
1answer
32 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
vote
0answers
39 views

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

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
31 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
53 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
24 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
vote
4answers
81 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.
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
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
79 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
vote
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 ...