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

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1answer
84 views

A counterexample to $Ae\simeq Ae^{\prime}\to A(1-e)\simeq A(1-e^{\prime})$

Let $e,e^{\prime}$ be two idempotents in a $k$- algebra $A$ ($k$ is a field) . Then my guess $Ae\simeq Ae^{\prime}$ (as a left $A$- module) does not imply $A(1-e)\simeq A(1-e^{\prime})$ in general, ...
6
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2answers
147 views

Product of Nilpotent ideal and simple module

I am stuck with trying to show that if an ideal I of a ring R is Nilpotent and M is a Simple R-Module, then IM = 0 I have attempted showing this by using the fact that the annihilator of a simple ...
3
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1answer
68 views

Finding simple modules in fields

I am struggling with finding all simple are modules for general rings i know that the simple modules of R correspond with the simple modules of R/rad(R), where rad(R) is the Jacobson radical but i ...
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1answer
33 views

On the construction of an $R$-Module

Let $X \neq \emptyset$ be a set and $(R,+, \cdot)$ a commutative Ring with $\mathbb{1}$ and $(N,+, \cdot)$ an $R$-Module. Show that $(\text{map}(X,N), +, \cdot)$ is an $R$-Module where for $A= ...
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1answer
31 views

Prove $Rm$ simple $\iff Ann(m)$ is a left ideal of $R$.

I'm trying to that prove $Rm$ is a simple ring $\iff Ann(m)$ is a left ideal of $R$. I did a couple of questions earlier that I believe could be useful to show this: $(Q1)$ Quotient of cyclic ...
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1answer
27 views

Functorial isomorphism involving tensor products

Let $R$ be a commutative ring and $E', E, F', F$ be free, f.g. $R$-modules of equal rank. For $f\in L(E',E):={\rm Hom}_R(E',E)$ and $g\in L(F',F)$, let $T(f.g)\in L(E'\otimes_R F', E\otimes_R F)$ be ...
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2answers
46 views

Prove isomorphism for a commutative ring

Show that $M\simeq \operatorname{Hom}_R(R,M)$ considering them as $R$-modules and $R\simeq \operatorname{Hom}_R(R,R)$ considering them as rings. I couldn't find a way to define the isomorphism.
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0answers
22 views

Modules over PIDs

Let $M=\mathbb{Z}^4/N$ where $N$ is a subgroup of $\mathbb{Z}^4$ generated by $(1,0,-1,3)$ and $(2,4,8,-6)$. Recognize $M$ as a product of cyclic groups. Here I have to use the following Theorem: If ...
3
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4answers
88 views

If every free $R$-module has the property that independence implies extendibility, is $R$ necessarily a field?

Definition. Whenever $M$ is a free $R$-module, let us call a subset $A$ of $M$ extendible iff there is a basis $B$ for $M$ such that $A \subseteq B$. (Is there a standard name for this condition?) ...
2
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1answer
60 views

Let $R$ be a $M\times N$ matrix with rational entries. Is $|(R\mathbb{Z}^N)/\mathbb{Z}^M|=|(R^T\mathbb{Z}^M)/\mathbb{Z}^N|$?

Let $R$ be a $M\times N$ matrix with rational entries, $R\mathbb{Z}^N$ be the image of $\mathbb{Z}^N$ under $R$. Consider an equivalence relation on $R\mathbb{Z}^N$ defined by $a\sim b$ if $a-b\in ...
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1answer
60 views

A simple module is necessarily the socle of its injective hull?

According to the wikipedia page on injective hulls "a simple module is necessarily the socle of its injective hull". It is clear to me that why a simple module is a subset to the socle of of its ...
1
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1answer
21 views

Intersection distributes over equality

Let $N$ be a submodule of an $R$-module $M$ and $I$ an ideal of $R$. Is the equality $$\frac{\bigcap_{n\ge 1}(I^nM+N)}{N}=\bigcap_{n\ge 1}(\frac{I^nM+N}{N})$$ true?
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1answer
47 views

Classification of separable algebras over a commutative ring

A separable algebra over a field can be classified as a finite product of matrix algebras over division algebras whose centers are finite dimensional separable field extensions of the field. (See ...
3
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1answer
61 views

Representatives of simple modules

For each of the following rings find a list of representatives of all simple $R$-modules: 1) $R=\mathbb C[x]$ 2) $R=\mathbb R[x]$ What I've tried was: I know that $M$ is a simple ...
1
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1answer
19 views

Boundary Homomorphism

I was studying the proposition 2.10 of Atiyah and MacDonald's Introduction to Commutative Algebra, and have a question. The proposition says: Let $$ \require{AMScd} \begin{CD} 0 @>>> ...
2
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2answers
65 views

If $\Gamma$ is $\Lambda$-projective and $C$ is $\Gamma$-injective, then $C$ is $\Lambda$-injective.

This is a problem I ran into while reading Cartan Eilenberg's Homological algebra pg 30. Given a unital ring homomorphism $\varphi:\Lambda\to\Gamma$, I want to prove the underlined statement. I ...
2
votes
1answer
40 views

Mitchell's Embedding Theorem for not-necessarily-small categories

Mitchell's Embedding Theorem states that if $\mathcal{A}$ is a small abelian category, then there is a ring $R$ and a fully-faithful exact functor $F:\mathcal{A}\rightarrow R\mathsf{Mod}$. To what ...
2
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1answer
69 views

Exact sequence of $A$-modules [duplicate]

I was trying to demonstrate the Proposition 2.9 of Atiyah and MacDonald's Introduction to Commutative Algebra. But I couldn't do the following: Let $M$, $M'$, and $M''$ be $A$-modules, $v$ and $u$ ...
0
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0answers
37 views

Surjectivity of a Free Module Homomorphism implies Injectivity?

I am trying to prove the following: Let $R$ be a commutative ring and $M$ be a free $R$-module having a finite basis of $n$ elements. Let $T:M\to M$ be a surjective $R$-module homomorphism. ...
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0answers
44 views

How to compute $Ext_A^{1}(S_1, S_2)$ and $Ext_A^{1}(S_2, S_1)$?

Let $A = kQ/\rho $, $Q$ is the quiver \begin{align} 1 \overset{a}{\underset{a^*}{\rightleftarrows}} 2 \end{align} $\rho$ is the relation $a a^* - a^* a = 0$. Question: compute $Ext_A^{1}(S_1, S_2)$. ...
3
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2answers
138 views

Atiyah and Macdonald, Proposition 2.9

The following simple claim is used without proof in Proposition 2.9 of Atiyah and MacDonald (p.23). Although I believe I can prove it with a fairly involved argument, the claim is treated by the ...
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1answer
173 views

Verifying statement of exact sequences

I'm trying to read through Atiyah and MacDonald's Introduction to Commutative Algebra. Proposition 2.9 says a sequence of $A$-modules and homomorphisms $$ M'\stackrel{u}{\to} M\stackrel{v}{\to} ...
0
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1answer
29 views

Direct summand of an injective module is injective

I am thinking the following statement is true: Direct summand of an injective module is injective Given $0\to A\to A'$ and $A\to C$, we compose the canonical injection to obtain $A\to C\oplus ...
3
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1answer
36 views

Is a stably free module always free? [duplicate]

Let $R$ be a commutative ring with unity and $M$ an $R$-module. If $M\oplus R^m\cong R^n$ for some integers $m,n \geq 1$ then must $M$ be finitely generated and free? Can somebody help me with ...
2
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0answers
58 views

Modules with finite injective dimension have $\omega_R$-resolutions

Let $(R,m,k)$ be a local Noetherian ring, $M$ a finitely generated $R$-module. How can I prove that $M$ has finite injective dimension if and only if it has a $\omega_R$-resolution? ($\omega_R$ is the ...
2
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2answers
37 views

Is the length of a module over a simple artinian ring an invariant?

If $R$ is a simple artinian ring, Wedderburn theory tells us that $R=Mat_n(D)$ for some $n\geq 1$ and division ring $D$ and also every $R$-module $M$ is a direct sum of finitely many copies of the ...
6
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2answers
71 views

Rank-nullity theorem for free $\mathbb Z$-modules

From linear algebra we know that given vector spaces $V$, $W$ over a field $k$ and a linear map $f\colon V\to W$ we have $$\dim V = \dim \operatorname{im} f + \dim \ker f.$$ Is this still true when ...
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1answer
31 views

Reference on Modules

I'm looking for a book that provides a nice introduction to Modules for a student that already had a first course in Abstract Algebra in groups and rings. The book should explain why are modules ...
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0answers
10 views

Multiplication Module

We say that $R$-module $M$ is a multiplication module, if for every submodule $N$ of $M$, there is an ideal $I$ of $R$ such that $N=IM$. Now, I have two questions: What is an example of $R$-module ...
0
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1answer
15 views

finding clusters in a network from eigengaps

I have a usual Laplacian matrix, which describes a network. From the matrix I get the eigenvalues and from these I can compute a metric of modularity in my network based on the largest eigengap. Let's ...
6
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2answers
118 views

${\rm Hom}_R(M, R/M) =\{0\} \implies R$ is a field.

Let $R$ be a local ring with maximal ideal $M$. Suppose $M$ is finitely generated. Prove that if ${\rm Hom}_R(M, R/M) =\{0\}$, then $R$ is a field. ${\rm Hom}_R(M, R/M)$ stand for the group of ...
5
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1answer
378 views

Prove $(2, x)$ is not a free $R$-module.

Let $R = \mathbb Z[x]$ and let $M = (2, x)$ be the ideal generated by $2$ and $x$, considered as an $R$-module. Show that $\{2, x\}$ is not a basis for $M$. Show that any 2 elements of ...
0
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1answer
21 views

Modules that allow an infinite exact sequence on them

I'm looking for a characterization of modules that fit the following property: (*) There's an infinite exact sequence with $\phi_i \neq 0 \space \forall i\in I $ $\require{AMScd}$ \begin{CD}\cdots ...
2
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1answer
58 views

Kernel of a retraction

I have a couple of questions about a exercise I have: Let $0\to L\stackrel{\alpha}\to M\stackrel{\beta}\to N \to 0$ be a splitting exact sequence and let $r$ be a retraction of $\alpha$ such that ...
4
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1answer
99 views

Rank of projective module defined as the smallest $n$ such that $P$ is a direct summand of $R^n$

Over a commutative ring $R$, the rank of a projective module $P$ is defined by looking at the map $\text{rank}(P) : \text{Spec}(R) \rightarrow \mathbb{N}_0$ given by $\mathfrak{p}\mapsto ...
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0answers
85 views

Prove that module has finitely many elements

Let $p$ be a prime number. Consider the subring $U:= \mathbb{Z}[1/p]$ of $\mathbb{Q}$ and define the $\mathbb{Z}$-module $M:=U/ \mathbb{Z}$ (1): Show that any $\mathbb{Z}$-submodule of $M$ that is ...
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1answer
58 views

Rank-nullity theorem for modules

Let $R$ be a ring, $M$ be a finitely generated free $R$-module and $f \in \text{End}_R(M)$. Does the following hold? $$\text{rk}(M)=\text{rk(Ker}(f))+\text{rk(Im}(f)).$$ This is the last question of ...
0
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1answer
15 views

Show that map from $\text{Hom$_A$($_AA,M$)} \to M$ is injective

Let $A$ be a unital ring and $M\in\text{A-Mod}$ Then $\text{Hom$_A$($_AA,M$)} \cong M$ By $\phi \mapsto \phi(1)$ I want to show that this map is injective, without using Yoneda Lemma. I have a proof ...
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2answers
101 views

If a module is nonzero, then a localization module is nonzero

Let $R$ be a commutative ring, when $\mathfrak p$ is a prime ideal, there is the localization $M_{\mathfrak p}:=S^{-1}M$, where $S=R\setminus\mathfrak p$. Show: If $M$ is a nonzero $R$-module, ...
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1answer
24 views

Confused about simple and semisimple modules

I was reading one of the possible definitions of semisimple modules, which is "for every submodule $N$ of $M$, there exists a submodule $P$ such that $M=N \bigoplus P$. After reading this I ...
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2answers
26 views

Correctness of the relation between free, torsion, torsion free, finitely generated module.

Is the following relation true? Is there a torsion module but not finitely generated module. (That is, an infinitely generated torsion module.) I am sure there is, because Here 2,(6) and Here ...
1
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1answer
26 views

Finite modules over (infinite) commutative rings

I'm attempting to solve the following two problems and I've unfortunately hit a wall. Q1. Show that if $M$ is a finite module over an infinite commutative ring $A$, then $M$ is a free module ...
3
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1answer
60 views

On regular elements and Maximal Cohen-Macaulay modules

I was reading theorem 3.3.3 in Bruns-Herzog: we have a Cohen-Macaulay local ring $(R,\mathfrak m,k)$, $C$ and $M$ are maximal Cohen-Macaulay modules. (Probably to solve my question some of these ...
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0answers
12 views

Dieudonné separation theorem for locally $L^{\infty} $-convex modules

Does anybody knows if there exists a version of the Dieudonné separation theorem for locally $L^{\infty} $-convex modules? I know that such a theorem exists for locally $L^{0} $-convex modules, but ...
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2answers
208 views

If the tensor product of two modules is free of finite rank, then the modules are finitely generated and projective

If over a commutative ring $R$ we have that $M\otimes N=R^n$, $n\neq 0$, need we have that $M$ and $N$ are finitely generated projective? We have finite generation, because if $M\otimes N$ is ...
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0answers
64 views

Basis of the ring $B=End_R(R^{(\mathbb N)})$

Let $B=End_R(R^{(\mathbb N)})$. Define $u,v \in B$ as $$u(e_{2_{i+1}})=0, u(e_{2_i})=e_i$$$$v(e_{2_{i+1}})=e_i,v(e_{2_i})=0$$ Prove that $\{u,v\}$ is a basis of $B$ as a $B$-module. I've already ...
0
votes
1answer
57 views

Does a homomorphism $x^k\mapsto k$ exist? [closed]

If $M=\mathbf{Z}[G]$ is the group considered as a module over itself, and with $\mathbf{Z}$-basis $\{1,x,\ldots,x^{n-1}\}$, is it possible to construct a module homomorphism $\varphi:M\to\mathbf{Z}$ ...
0
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0answers
17 views

Is this map $\mathbf{Z}$-bilinear?

Let $M=\mathbf{Z}[C_n]$ have a $\mathbf{Z}$-basis $\{e^1,\ldots,e^{2n-1}\}$, then I want to show the following map is $\mathbf{Z}$-bilinear. $\varphi:M\times M\to\mathbf{Z}$ ...
0
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0answers
22 views

When is the rank of a general tensor 1?

Suppose we have two modules $M$,$N$ with $\mathbf{Z}$-bases, and take the tensor product $M\otimes_\mathbf{Z} N$. If I use Bourbaki's definition that the rank of a general tensor T is defined to be ...
0
votes
2answers
48 views

If $R{^m}$ is isomorphic to $R{^n}$ as $R$-modules and if $M$ is a maximal ideal of $R$ then how can I show that image of $M{^m}$ is $M{^n}$?

If $R{^m}$ is isomorphic to $R{^n}$ as $R$-modules and if $M$ is a maximal ideal of $R$ then how can I show that image of $M{^m}$ is $M{^n}$? Background: I was trying to prove that if $R{^m}$ is ...