Questions tagged [modules]
For questions about modules over rings, concerning either their properties in general or regarding specific cases.
5,753 questions
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Trivial module functor is $\text{Hom}_\Bbb Z(\Bbb Z,-)$
Weibel - Homological Algebra - Page 161
Considering $\Bbb Z$ as a $\Bbb Z-\Bbb ZG$-bimodule, the "trivial module functor" from $\Bbb Z$-mod to $\Bbb ZG$-mod is the functor $\text{Hom}_\Bbb Z(\Bbb Z,...
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18 views
about direct sum decomposition of module
I know that M is free R module and $M_F$ free F module. I also know that how $M/M\cap N$ is free R module. I want to know why that $gs=id$ is identity map and how it yields direct sum decomposition of ...
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15 views
Another question on ideal and tensor product
Let $R$ be a commutative Noetherian ring and $M$ be an $R$-module. Let $I$ be a proper ideal of $R$ and $a,b \in R$ be such that $ab\in I$ and the maps $(I+Rb) \otimes_R M \to R\otimes_R M$ and $(I+ ...
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12 views
On ideal and tensor product , retaining injectivity
Let $R$ be a commutative Noetherian ring and $M$ be an $R$-module. Let $I$ be a proper ideal of $R$ and $b\notin I$ be such that the maps $(I+Rb) \otimes_R M \to R\otimes_R M$ and $(I:b) \otimes_R M \...
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0answers
24 views
When $P\otimes_R M \to R \otimes_R M$ is injective for every (finitely generated) prime ideals $P$ of $R$?
Let $R$ be a commutative ring. If $M$ is an $R$-module such that for every finitely generated prime ideal $P$ of $R$, the map $i\otimes Id: P\otimes_R M \to R \otimes_R M\cong M$ is injective, then ...
1
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0answers
21 views
commutative Noetherian ring whose every maximal ideal is projective
Let $R$ be a commutative Noetherian ring. If every maximal ideal of $R$ is projective as an $R$-module, then is $R$ hereditary?
3
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1answer
45 views
Sub-$\mathbb{Z}$-modules of $\mathbb{Q}$
Let $p,q\in\mathbb{N}$, with $p\neq q$, prime numbers and consider the sets:
$$H_{p}=\biggl\{\frac{a}{p^{k}}: a\in\mathbb{Z}, k\in\mathbb{N}\cup\{0\}\biggr\}$$
$$H_{q}=\biggl\{\frac{a}{q^{k}}: a\in\...
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1answer
9 views
Submodules of dyadic rationals
Let $D = \{j/2^n \mid j \in \mathbb{Z}, n \geq 0\}$ the set of dyadic rationals as a $\mathbb{Z}-$module. If $X = \frac{D}{\mathbb{Z}},$ what about the submodules of $X?$ There is any infinite ...
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4answers
43 views
If $r$ is a nilpotent element of a commutative ring $R$ then $(r)$ is not a direct summand of $R$ as an $R$-module.
I am having trouble proving that if $r$ is a nilpotent element of a commutative ring $R$ then $(r)$ is not a direct summand of $R$ as an $R$-module. I know that $(r)$ is then a nilpotent ideal. I ...
2
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1answer
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Step to prove that if |G| is infinite or |G| is divisible by char(k) then k[G] is not semisimple.
I know that there are plenty of resources proving that if |G| is infinite or |G| is divisible by char(k) then k[G] is not semisimple, but I was instructed to try it in a particular way.
I am down to ...
3
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1answer
50 views
Action of sum of traspositions on a simple $\mathbb{C}[S_n]$-module
Let $\lambda$ be a Young tableaux and $V_{\lambda}$ the standard simple $\mathbb{C}[S_n]$-module associated to it, constructed as $\mathbb{C}[S_n]c_{\lambda}$.
We denote by $x=\sum\limits_{1\leq i &...
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1answer
26 views
Show that the ring of upper triangular 2x2 matrices is a direct sum of two of its modules
The definition I have for the direct sum of R-modules is the following:
$\bigoplus_{i \in I} M_i = \big\{f \in \Pi_{i\in I}M_i : \ \#\{i \in I:\ f(i) \neq 0 \} < \infty \big\}$.
Say k is some ...
1
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0answers
18 views
When does deleting the $n$-torsion elements make no difference?
Given an abelian group $X$, there's a correspond $n$-torsion subgroup defined as follows: $X[n] = \{x \in X: nx = 0\}.$ Call a pair $(X,n)$ generative iff $\mathbb{Z}\langle X[n] \setminus X\rangle = ...
3
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1answer
45 views
Show this representation of $\mathfrak{sl}_2(\mathbb C)$ is completely reducible
Consider the homomorphism $\phi : \mathfrak{sl}_2(\mathbb C) \rightarrow \mathfrak{sl}_3(\mathbb C)$ sending $\left(\begin{matrix} a&b \\ c&d \end{matrix}\right)$ to $\left(\begin{matrix} a&...
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0answers
24 views
If $I$ is a left ideal of $R$ and $y$ is any arbitrary element in $R$. Then prove that for any $x\in R$, $xy\in I$ implies that $x\in I$ [closed]
Let $I$ be a left ideal of a non commutative ring $R$ and let $x,y\in R$. If $y$ is an element that is arbitrary in $R$ such that either $y\in I$ or $y\notin I$. Then prove that $xy\in I$ implies ...
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2answers
32 views
How to find the modules of a big number with a big powe ?? [closed]
How to find the modulus of a big number with a big power?
Such as $2222^{5555}$ or $5555^{2222}$ (mod 7)?
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0answers
23 views
Examples of modules over a group ring
I am in search of some examples of finite groups $G$ and modules over the ring $\mathbb{Z}[G]$ with following conditions (and examples taken independently). I do not have idea whether such examples ...
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0answers
18 views
Injective and Projective Modules
There are several equivalent definitions of projective and injective modules over a ring $R$ (with unity).
However, I didn't find anywhere the justification for using words injective and projective ...
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0answers
15 views
On cancelling primary ideals in Noetherian, local, unique factorization domain of dimension $2$
Let $(R,\mathfrak m)$ be a local , Noetherian , UFD of dimension $2$. Let $J$ be an $\mathfrak m$-primary ideal ($\sqrt J=\mathfrak m$) of $R$ such that $J^2=\mathfrak m J$.
Then, is it true that $J^...
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14 views
What is the compositum product module $MN$?
Proposition 5. Let $A$ be a ring contained in a field $L$. Let $В$ be the set
of elements of $L$ which are integral over $A$. Then $В$ is a ring, called
the integral closure of $A$ in $L$.
...
5
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1answer
84 views
+50
Ideal $I$ in a Noetherian ring such that $I+(b)$ and $I+(a)$ are projective, where $ab\in I$
Let $R$ be a Noetherian ring and $I$ be a proper ideal of $R$ and let $a,b\in R$ be such that $ab\in I$ and $I+Ra$ and $I+Rb$ are projective as $R$-modules.
Then is the ideal $I$ projective as an $R$...
1
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1answer
64 views
On projective ideals
Let $R$ be a Noetherian ring and $I$ be a proper ideal of $R$ and let $b\notin I$ be such that $I+Rb$ and $(I:b)=\{x\in R: xb\in I\}$ are projective as $R$-modules.
Then is the ideal $I$ projective ...
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0answers
17 views
reflexive ideal in regular local ring
Let $I$ be an ideal of a regular local ring $R$ such that $I$ is reflexive as an $R$-module (https://stacks.math.columbia.edu/tag/0AUY) .
Then is $I$ a principal ideal (i.e. a free $R$-module i.e. a ...
1
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1answer
9 views
Comparing left principal ideals in a non commutative ring with unity
If $R$ is a non commutative ring with unity. Then $Ra, Rba$ and $Rab$ are left principal ideals for some $a,b\in R$. What is the relationship between these left ideals? For instance, which one of ...
3
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1answer
38 views
Is the “basis” of a free module linearly independent?
I am reading Dummit and Foote (Section 10.3):
Definition: An $R-$ module $F$ is said to be free on the subset $A$ of $F$ if for every non-zero $x$ of $F$, there exist unique non-zero elements $r_1, ...
3
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1answer
43 views
Is $Hom_R(k^2,k^2)=k$?
Let $k$ be any field, and $R = Mat_{2 \times2}(k)$.
Is it true that $Hom_R(k^2,k^2)=k$?
I assume the question is asking if the two are isomorphic as $k$-vector spaces. Is this an example of Schur's ...
1
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1answer
44 views
Prove that any non zero $\mathbb{Z}$-module M has a nonzero homomorphism to $\mathbb{Q}/\mathbb{Z}$
I have not learned anything about 'injective objects', though I got familiar with divisible groups while I was trying to understading Let $G$ be any abelian group and $a\in{G}$. Show there exists a ...
4
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1answer
58 views
Is $R$ finitely generated?
Let $A$ be a commutative ring with identity. Given two submodules $R,S$ of $A^n(n\in\Bbb N)$ and suppose $S$ is finitely generated, if there exists an isomorphism of $A$-modules $A^n/R\simeq A^n/S$, ...
3
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1answer
33 views
Is faithful flatness implying $M \otimes_{S^{-1}R} S^{-1}N \cong M \otimes_R N$? [duplicate]
Assume that $M$ is faithfully flat $S^{-1}R$-module, for $R$ commutative ring and $S \subset R$ a multiplicative subset. Furthermore, let $N$ denote an $R$-module.
Is it true that $M \otimes_{S^{-1}...
2
votes
2answers
68 views
Do we have $R\simeq S$ for two submodules $R,S$ of $A^n$? [closed]
Let $A$ be a commutative ring with identity. Given two submodules $R,S$ of $A^n(n\in\Bbb N)$, if there exists an isomorphism of $A$-modules $A^n/R\simeq A^n/S$, then do we have $R\simeq S$?
0
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1answer
19 views
Why does every module over an $R$ conmutative ring is free, if $R$ is a field?
I know that the condition is an "if and only if". I've proved that if every $R$ module is free then $R$ is a field, but I can't see so clearly the converse.
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0answers
18 views
Do we have an isomorphism of $A$-modules $R\simeq S$? [duplicate]
Let $A$ be a commutative ring with identity and $M$ an $A$-module. Given two submodules $R,S$ of $M$, if there exists an isomorphism of $A$-modules $M/R\simeq M/S$, then do we have $R\simeq S$?
0
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1answer
28 views
Problem on the Fitting ideal
I am trying to solve Dummit Foote, Abstract Algebra, 15.1 Exercise 40. It reads,
Suppose $R$ and $S$ are commutative rings, $\phi:R \to S$ is a ring homomorphism, $M$ is a finitely generated $R$-...
1
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1answer
27 views
Module of eventually zero sequences over a Jacobson algebra is simple
Consider the Jacobson algebra $R =k \langle x, y\rangle /(xy − 1)$, where $k$ is a field. Let $k^ω$ be
the R-module whose elements are infinite sequences in $k$, with $y$ acting as the right
shift ...
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0answers
28 views
Map $f:M \to M^{\vee \vee}$ to Double Dual Space
Let $R$ be a ring and $M$ a coherent $R$-module, therefore there exist a finite presentation: an exact sequence
$$R^I \to R^J \to M \to 0$$ (*)
with finite index sets $I,J$.
Let consider the ...
1
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1answer
27 views
When does a ring homomorphism yield module structure?
Assume all rings have 1, preserved by homomorphisms, and that all modules are unitary.
Given a ring homomorphism $\phi:R\to S$, any $S$-module A can be given an $R$-module structure by the action $ra\...
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0answers
41 views
Surjective module homomorphism $R^n \twoheadrightarrow R^n$ that is not an isomorphism [duplicate]
Let $R$ be a commutative ring with unit, and $f: R^n \twoheadrightarrow R^n$ be a surjective $R$-module homomorphism. I was able to show that $f$ is an isomorphism if $R$ is a local ring by looking at ...
0
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1answer
16 views
Let $M=\Bbb Z^2$ . If $N$ and $P$ internal direct summands of $M$ , then does it follow that $N+P$ is also an internal direct summand of $M$
Let $M=\Bbb Z^2$ . If $N$ and $P$ internal direct summands of $M$ , then does it follow that $N+P$ is also an internal direct summand of $M$.
My attempt :
$\Bbb Z^2=\Bbb Z \oplus \Bbb Z$. Then $N=P=\...
2
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0answers
23 views
Give an example where $Ann(N) +Ann(K) \ne Ann(N\cap K)$
Let $N$ and $K$ be submodules of an $R$-module $M$, with $Ann(N)$ and $Ann(K)$. Give an example where $Ann(N) +Ann(K) \ne Ann(N\cap K)$
My attempt :
We consider $M=\Bbb R^2$ as an $\Bbb R$-module. ...
1
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0answers
40 views
If $M = P \oplus Q$ then is $M\cap N = (P \cap N) \oplus (Q \cap N)$
(1) If $M,N,P,Q$ are $R$-modules and $M = P \oplus Q$ then is $M\cap N = (P \cap N) \oplus (Q \cap N)?$
Since the modules are not said to be anything like finitely generated or free, I really don't ...
3
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3answers
53 views
Is $\Bbb Q$ a decomposable module over $\Bbb Z$ or not?
Is $\Bbb Q$ a decomposable module over $\Bbb Z$ or not?
My attempt:
let, $p_1,p_2,\dots,p_k,\dots$ be an enumeration of primes in $\Bbb N$. Then, can't we write $\Bbb Q = \Bbb Z \oplus Z(\frac{1}{...
2
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2answers
34 views
Show that $\ker(\phi)$ is finitely generated
$\DeclareMathOperator\Ker{Ker}$Let $\phi : M \to F$ be a surjective homomorphism of a finitely generated module $M$ onto a free module $F$. Show that $\Ker(\phi)$ is finitely generated.
My attempt :
...
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0answers
22 views
Is an infinite direct sum of modules still a product? Coproduct?
I know nothing about categories, but today I did an exercise proving that for a module $M$ :
$M, \xi_1,\dots, \xi_d$ is a product $\iff M \cong \bigoplus_{i=1}^d M_i \iff$ $M, \alpha_1,\dots,\alpha_d$...
0
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0answers
58 views
$M/PM$ is torsion-free
Let $R$ be a PID and $S$ be a finite ring extension of $R$ which is a free module over $R$. Moreover, let $S$ be reduced and of dimension one. Let $M$ be a finitely generated $S$-module which is ...
3
votes
1answer
25 views
Is there a set of isomorphism classes of indecomposable $R$-modules?
Let $R$ be an associative ring with one.
Is there a cardinal $\kappa$ such that each indecomposable $R$-module has cardinality less than $\kappa\ ?$
Equivalently, is there a set $S$ of ...
3
votes
1answer
37 views
Every monic is a kernel
This is part of Weibel's Exercise 1.2.2, where I have to show that in the category R-Mod, every monic is a kernel.
A monic morphism is defined to be a map $i \colon A \to B$ such that if $g \colon A ...
7
votes
2answers
98 views
Is the sheafication of a “presheaf of $\mathcal{O}_X$-modules” an $\mathcal{O}_X$-module?
Let $(X,\mathcal{O}_X$) be a ringed space. A presheaf of $\mathcal{O}_X$-modules is a presheaf $\mathcal{F}$ of abelian groups on $X$ such that $\mathcal{F}(U)$ is an $\mathcal{O}_X(U)$-module for ...
5
votes
1answer
66 views
$R$ is a unital commutative ring, $M$ and $N$ are $R$-modules. If $f:M \to N$ is $R$-linear, then is it true that $M= \ker(f) \oplus \text{im}(f)$?
Let $R$ be a commutative ring with unity. Prove or disprove: for $R$-modules $M$ and $N$, if $f:M \to N$ is $R$-linear, then $M= \ker(f) \oplus \operatorname{im}(f)$.
My attempt:
Let $f : \Bbb Z/4\...
1
vote
1answer
23 views
example to show that an increasing union of finitely generated submodules of some module need not be finitely generated
Give an example to show that an increasing union of finitely generated submodules of some module need not be finitely generated.
My attempt:
Let $R = \Bbb Z$ .
Consider $M=\Bbb Z \oplus \Bbb Z(\...
0
votes
1answer
49 views
How to show $\operatorname{Hom}_A(M,N)$ is a finitely generated $A$-module?
Let $A$ be a commutative Noetherian ring with identity and $M,N$ two finitely generated $A$-modules.
How to show $\operatorname{Hom}_A(M,N)$ is a finitely generated $A$-module?
