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

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4
votes
1answer
217 views

How does one show that this tensor product is not torsion-free?

I am having trouble showing that a particular tensor product is not torsion-free. Let $ R = k[[x,y]] $, where $ k $ is a field (this is the ring of formal power series in $ x $ and $ y $ with ...
10
votes
0answers
223 views

Non-reflexive module isomorphic to its double dual

Could you give me an example of a non-reflexive module isomorphic to its double dual? I found an example here but I cannot understand it, do you have any simpler examples? By this question we ...
3
votes
1answer
742 views

Proving that Hom$(N,-)$ is left exact.

Say we have a ring $R$ and let $A,B,C$ be $R$-mod. Then prove $\hom_R(N,-)$ is left exact (where $N$ is some fixed $R$-mod). Basically we want to show that given that we know that $0\rightarrow ...
2
votes
1answer
302 views

tensor product over local ring

If $R$ is a local ring and $M$ and $N$ are finitely generated R-modules such that $M\otimes N = 0$ then it follows from Nakayama's lemma that either $M=0$ or $N=0$. This I know. But now I am looking ...
0
votes
1answer
258 views

Module Homomorphisms and a Direct Sum

I am trying to show the following: Let $f:M\to N$ and $g:N\to M$ be module homomorphisms such that $g\circ f=Id_M$. Prove that $N=Im(f)\oplus\ker(g)$. I know that $M/\ker(f)\cong Im(f)$, but I'm not ...
3
votes
1answer
449 views

Using Nakayama lemma to prove that surjective implies injective.

$A$ is a commutative ring, and $f:M\rightarrow M$ is an endomorphism of $A$-modules which is surjective. If I know that $M$ is finitely generated, I want to prove that $f$ is also injective. ...
2
votes
1answer
75 views

number of possible module structures on this abelian group.

So I am being asked how many $\mathbb{Z}[x]$ module structures does the abelian group $\mathbb{Z}/5\mathbb{Z}$ have. Thinking about it, we really only need to define the action of $x$ on $[1]$ ...
4
votes
1answer
82 views

Find $n$ such that $\mathbb{Z}/n\mathbb{Z}$ has a $\mathbb{Z}[i]$-module structure.

We want to find all $n>0$ such that $\mathbb{Z}/n\mathbb{Z}$ has a $\mathbb{Z}[i]$-module structure. This is what I have. First of all we have that for $k\in \mathbb{Z}$ we have that for ...
4
votes
1answer
107 views

Is the section of a surjective endomorphism again a morphism?

Let $R$ be a commutative ring with unit and let $M$ be an $R$-module (with $1\cdot m = m$ for all $m\in M$). Let $f:M\twoheadrightarrow M$ be a surjective morphism of $R$-modules of $M$ onto itself. ...
3
votes
2answers
67 views

Some questions of PIDs

I'm stuck on a couple of practice problems relating to PIDs, they are paraphrased below: Given a PID $R$ with $a$ and $b$ in $R$ and gcd$(a,b)=1$ I need to show that: 1) There are elements $s$ and ...
3
votes
2answers
175 views

Self-injective noetherian rings

I would like to prove that the following conditions for a ring $R$ are equivalent: 1) $R$ is Noetherian and self-injective; 2) The class of projective $R$-modules is equal to the class of ...
1
vote
1answer
202 views

Finitely generated free modules of infinite rank.

We know that for general modules over a commutative ring with $1$, you can't always extract a basis from a generating set. This makes me think that maybe there should be free modules of infinite ...
1
vote
1answer
65 views

Indecomposable L-module

I have the following exercice which I have be trying to solve: Let L be a Lie algebra and $r:L\rightarrow gl_3(F)$ a representation of L such that $im(r)=t_3(F)$ (the upper triangular matrices). Show ...
3
votes
1answer
116 views

Homomorphism from module to localization

I have a question about this statement: Let $M$ be an $R$-module of finite length and let $\mbox{Ann}(M)\subset P_1,...,P_k$ be maximal ideals. If $n\in\mathbb{Z}_+$ is such that $P_1^n\cdots ...
1
vote
1answer
199 views

further question on the existence of decomposable modules

This is related to another solved question submodule of indecomposable module It has shown some indecomposable modules have decomposable modules. Now my question is, Suppose M is an injective and ...
3
votes
0answers
455 views

Structure theorem of finitely generated modules over a PID

I want to prove the structure theorem of finitely generated modules over a PID using the primary decomposition in a Noetherian $R$-module $M$. Applying the results on primary decomposition to the ...
3
votes
1answer
85 views

Question dealing with modules and socles.

To help myself prepare for an upcoming exam, I've been working on various problems that focus on the topics we have been dealing with in class. I came across this one: Let $M$ be a right ...
2
votes
1answer
78 views

Composition length of surjective inverse limits

Let $M$ be a left module over some ring $R$ and suppose that $M$ is an inverse limit of a family of modules $M_i$ with $i\in I$. We suppose also that the maps of the inverse limit $\pi_i:M\to M_i$ are ...
0
votes
1answer
80 views

What is a good way to prove a module is finitely generated if and only if the union of every chain of proper submodules of $M$ is a proper submodule?

The take-home exam give me the following problem: Prove that a module is finitely generated if and only if the union of every chain of proper submodules of $M$ is a proper submodule. For the if ...
1
vote
2answers
771 views

Is every finitely generated module Noetherian?

I used this "fact" when I hand in my take home exam. To my surprise this was returned with a remark "$M$ is not noetherian!". But I remember the criterion for noetherian is $M$'s submodules are all ...
8
votes
2answers
365 views

Seeking an example in module theory — (In)decomposable modules

Can anyone give me a module $M$ over a ring $R$, such that $M$ is indecomposable, but $M$ has a submodule $N$ such that $N$ is decomposable? In general, what further assumptions can we put on the ...
2
votes
3answers
119 views

Why is $\mathbb{Z}/n\mathbb{Z}$ quasi-Frobenius?

When surfing the wiki, I found the definition of Quasi-Frobenius rings $R$ is quasi-Frobenius if and only it satisfies the following equivalent conditions: All right (or all left) R modules which ...
5
votes
2answers
242 views

Isomorphic abelian groups which are not isomorphic R-modules

I'm looking for an example of of two isomorphic abelian groups, which are not isomorphic $R$ modules for some ring $R$. I suppose we can just the same abelian group $M$ twice, and use a different ...
-2
votes
2answers
470 views

Noetherian prime ideals and Noetherian ring

If $R$ is a commutative ring, and for every $P$ a prime ideal of $R$, $P$ is a Noetherian $R$-module, show that $R$ is Noetherian.
3
votes
1answer
72 views

Stabilizer of $\operatorname{Aut}(B)$ acting on $0\to A\rightarrow B\rightarrow C\rightarrow 0$

I am reading an article on elementary homological algebra and have a trouble understanding one statement. Let $R$ be a ring and $A,B,C$ modules over $R$. Let $S$ be a set of exact sequences of the ...
2
votes
1answer
87 views

$A/I$ is an injective $A/I$-module, where $A$ is PID

Suppose $A$ is PID, $I\subset A$ is a nonzero ideal, show $A/I$ is an injective $A/I$-module. I tried to prove this, but got stuck in my following argument. To show $A/I$ is injective as ...
3
votes
0answers
101 views

On complexes of projective modules

How can I prove the following statement? Let $\beta: B\rightarrow C$ be a quasi-isomorphism of complexes of $R$-modules. If $P$ is a complex of projective $R$-modules which is bounded below, then ...
2
votes
1answer
350 views

Extension of scalars

Let $A\rightarrow B$ be a ring homomorphism, $M$ and $N$ - modules over $A$. How to prove, that $$ (M \otimes_A N) \otimes _A B = (M \otimes_A B) \otimes_B (N\otimes_A B) $$ as $B$-modules in the ...
9
votes
2answers
340 views

The Noether-Deuring Theorem

I have to solve the following exercise taken from the book "Introduction to Representation Theory" by P. Etingof, O. Golberg, S. Hensel, T. Liu, A. Schwendner, D. Vaintrob, E. Yudovina and S. ...
0
votes
1answer
47 views

How to solve the following special inequality?

Find $k$, as a function of $d_2$ and $d_3$, such that: $$\left \vert { d_2 \left [ \sin(e^{d_3\,y}) - \sin(e^{d_3\,x})\right] + (x-y) d_2 d_3 e^{d_3\,z} \cos(e^{d_3\,z})} \right \vert \le k ...
0
votes
1answer
34 views

Free Fractional Modules

Let $R$ be a Dedekind domain with fractional ideal $M$. I am trying to prove: If $M$ is free, then the ideal $I$ such that $M=d^{-1}I$, where $d \in R$, is principal. Since $R$ is a Dedekind ...
4
votes
1answer
176 views

Chinese remainder theorem and isomorphism

Suppose that there are different primes $p_1$ and $p_2$ and the group $C_{p_1}$. I would like to decompose the following tensorproduct and thought of using the chinese remainder theorem to get ...
1
vote
1answer
93 views

$G/G' \cong I/I^2$ where $I$ is the augmentation ideal [duplicate]

Possible Duplicate: Isomorphism between $I_G/I_G^2$ and $G/G'$ Let $G$ be a finite group. Let $I\unlhd\mathbb{Z}[G]$ be the augmentation ideal of the integral group ring $\mathbb{Z}[G]$. ...
1
vote
1answer
397 views

Hungerford Algebra - Chapter IV - proof of the splitting lemma

Im working with the splitting lemma right now and I don't understand it too much. If someone has that book and could help me complete the sketch in page 177, proof of 1.18) I would be very grateful. ...
2
votes
1answer
72 views

The indecomposable projective $\mathbb{F}_pG$-module with $UJ/J\cong \mathbb{F}_p$

Let: $G$ be a finite group; $p$ be prime; $J$ be the Jacobson radical of $\mathbb{F}_pG$. A paper I'm trying to read mentions the following object: The indecomposable projective ...
0
votes
1answer
32 views

$F(M)$ as a $\text{Hom}_{R_1}(M,M)$-module for some functor $F:R_1-\text{mod}\rightarrow R_2-\text{mod}$

I'm trying to generalize one aspect of the accepted answer in: $\text{Hom}(\mathbb{F}_p G, M)$ and $H^1(G,M)$ Is the following statement correct? If: $R_1$ and $R_2$ are rings; ...
0
votes
2answers
151 views

How to compute $\mathbb{Z}/m\mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Z}/n\mathbb{Z}$? [duplicate]

Possible Duplicate: Proof of $(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z}) \cong \mathbb{Z}/ \gcd(m,n)\mathbb{Z}$ How to compute $\mathbb{Z}/m\mathbb{Z} ...
4
votes
1answer
158 views

Divisible module proof: $\mbox{Ext}(R/aR; M) = 0, \forall a \in R \Rightarrow M $is divisible

This is a problem from my lecture notes, and I don't really know how to tackle it. Any help will be appreciated. The question reads as follow: Problem Given an integral domain $R$, and an ...
1
vote
1answer
149 views

A Gorenstein domain that is not a complete intersection

Could you give me an example (with proof) of a Gorenstein domain that is not a complete intersection?
1
vote
1answer
83 views

Why the depth of this modules is 1?

I'm studying on these notes, my question in about a proof on page 63. Basically this is my question: Suppose $R$ local noetherian of positive depth, $M$ a module of finite gorenstein dimension ...
19
votes
2answers
2k views

Applications of the Jordan-Hölder Theorem.

All books about group theory bring the Jordan-Hölder theorem, but does not give applications. I only saw in Rotman's book the Fundamental Theorem of Arithmetic being proved as a consequence of this ...
2
votes
1answer
119 views

When is a ring considered as a module over a Noetherian subring itself Noetherian?

If $S$ is a ring and $R$ is a Noetherian subring of $S$, and we know $_RS$ (i.e. $S$ viewed as a left $R$-module) is finitely generated (hence Noetherian), is $_SS$ necessarily Noetherian? I can't ...
1
vote
2answers
784 views

Finitely generated module in exact sequence

For $A$-modules and homomorphisms $0\to M'\stackrel{u}{\to}M\stackrel{v}{\to}M''\to 0$ is exact. Prove if $M'$ and $M''$ are fintely generated then $M$ is finitely generated.
5
votes
1answer
97 views

The Auslander dual commutes with ring extensions

Suppose $R$ noetherian and $M$ a finitely generated $R$-module. If you have a projective presentation of $M$: $P_1\rightarrow P_0\rightarrow M\rightarrow 0$, then by dualizing you obtain the following ...
3
votes
1answer
50 views

Relating Modules and Representations

I am currently studying representation theory and am struggling with the concepts which relate modules and representations. The specific question I am looking at right now is this: but while ...
2
votes
1answer
99 views

Counterexample for ${\rm Hom}_{R}(\prod_{i\in I}M_{i},N)\cong_{\mathbb{Z}} \coprod_{i\in I}{\rm Hom}_{R}(M_{i},N)$.

Let $\{M_{i}\}_{i\in I}$ be a family of $R$-modules and also $N$ is a $R$-module. Is there an counterexample for the following relation: ${\rm Hom}_{R}(\prod_{i\in I}M_{i},N)\cong_{\mathbb{Z}} ...
6
votes
1answer
657 views

When $\operatorname{Hom}_{R}(M,N)$ is finitely generated as $\mathbb Z$-module or $R$-module?

Assume that $M$ and $N$ are two finitely generated $R$-modules. Then $\operatorname{Hom}_{R}(M,N)$ is a finitely generated $\mathbb Z$-module and/or $R$-module (in this case, assume that $R$ is ...
1
vote
1answer
268 views

Exact sequences of Modules

For $A$-modules and homomorphisms $0\to M′\stackrel{u}{\to}M\stackrel{v}{\to}M′′\to 0$ is exact iff for all A-modules N, the sequence $0 \to Hom(M′′,N)\stackrel{\bar{v}}{\to} ...
3
votes
3answers
321 views

Error in proof that submodules of f.g. modules are f.g.

All modules are over a ring $R$ which is a PID. I wanted to show that every submodule of a finitely generated module is finitely generated. A key to this problem seems to be the following theorem (not ...
1
vote
0answers
93 views

Characterization of Hilbert schemes of points

Let $X=\operatorname{Proj}(R)$ be a projective scheme with a $k$-algebra $R$. Let $\mathcal{M} \in \operatorname{Coh}(X)$ be a coherent sheaf on $X$ whose Hilbert polynomial $\chi(\mathcal{M})$ is a ...