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Questions tagged [modules]

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

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1answer
18 views

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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0answers
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 ...
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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?
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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
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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 ...
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1answer
12 views

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 ...
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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 = ...
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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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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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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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0answers
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 $L$. ...
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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 $R$...
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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 ...
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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}...
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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 $R\simeq S$?
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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 $R\simeq S$?
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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 $R$-...
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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 ...
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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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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 ...
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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=\...
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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. ...
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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 ...
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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}{...
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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 $\iff M \cong \bigoplus_{i=1}^d M_i \iff$ $M, \alpha_1,\dots,\alpha_d$...
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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
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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
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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
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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
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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
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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
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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?