Tagged Questions

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

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0
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0answers
12 views

Irruducible $R$-modules of a ring [on hold]

Suppose $R$ is a ring and $M$ is an $R-$module. If $M$ is irreducible, then what is the meaning of it?
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1answer
37 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 ...
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0answers
42 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 ...
0
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1answer
14 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 ...
1
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1answer
17 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
18 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
24 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 ...
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0answers
8 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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1answer
40 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 ...
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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. $\varphi:M\times M\to\mathbf{Z}$ ...
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2answers
96 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
50 views

Does a homomorphism $x^k\mapsto k$ exist?

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 $\varphi:M\to\mathbf{Z}$ ...
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0answers
15 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 ...
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2answers
39 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 ...
2
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2answers
29 views

Semisimple modules and the radical

I don't need a proof, but can someone tell me whether it is true that for all $A$-modules $V$ we have that $V/\text{rad}V $ is semisimple, where we define $\text{rad} V$ as the intersection of all ...
1
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1answer
27 views

A property about quasi-primary modules

It is a fact that any discrete valuation domain $R$ has the property "P" that any proper submodule $N$ of any $R$-module $M$ is quasi-primary, in the sense that $\operatorname{rad}(N:M)$ is a prime ...
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1answer
30 views

Equivalent definitions of an algebra over a ring

I have seen two different definitions from several sources of an algebra over a ring. From Wikipedia: Let R be a commutative ring. An R-algebra is an R-Module A together with a binary operation ...
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0answers
48 views
+50

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 ...
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1answer
40 views

Can you always construct a map $A\otimes B\to A\times B$?

Suppose we have two $R$-mods $A,\,B$. Can we always construct a homomorphism $A\otimes B\to A\times B$?
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0answers
109 views

Localization of modules and primary decomposition

Let $(R,\mathfrak m)$ be a Noetherian local ring and $M$ be an $R$-module. Suppose that for all prime ideals $\mathfrak p$, $\mathfrak p\ne\mathfrak m$, the localization $M_{\mathfrak p}$ is the ...
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1answer
21 views

Question on Wedderburn components of $\mathbb C[G]$

$G$ is a finite group. Wedderburn's theorem says that $\mathbb{C}G\cong R_1 \times\cdot \cdot \cdot R_r$ as rings where $R_i=M_{n_i}(D^i)$ is the ring of $n_i\times n_i$ matrices over a division ring ...
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1answer
34 views

$(2,1+\sqrt{-5}), (1-\sqrt{-5},2)$ generate the $\mathbb Z[\sqrt{-5}]$-module $\langle 2,1+\sqrt{-5} \rangle \times \langle 2,1+\sqrt{-5} \rangle$

Ok, boring question here (I guess, at least). Let $R=\mathbb Z[\sqrt{-5}]$. Let $M=\langle 2,1+\sqrt{-5} \rangle$ the $R$-module generated by $2$ and $1+\sqrt{-5}$. I am asked to show that $M \times M ...
0
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1answer
13 views

Area between $y=2+|x-1|$ and $y=-\frac{1}{5}x+7$

Question 17, page 448, from Anton 8th. The question asks for the area, and the answer is 24. Now, I did draw the graph, found the points where both functions touch each other by $f(x) = g(x)$: ...
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0answers
17 views

direct sum of modules and generator subset

I am trying to solve the following problem: Let $(M_i)_{i \in I}$ be an infinite family of non zero modules and $S$ a system of generators of $\bigoplus_{i \in I}M_i$. Prove that the cardinal of $S$ ...
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1answer
24 views

Characterize semisimple rings with a unique maximal ideal

Problem Characterize the semisimple rings $R$ that contain a unique maximal ideal. I am not so sure what to do here. I know that a ring $R$ is semisimple if and only if all $R$-modules are ...
1
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1answer
26 views

Prove or disprove statements about modules

I am trying to determine if the following statements are true or false (i) There are free modules with non zero elements $x$ such that $\{x\}$ is linearly dependent. (ii) There are non free modules ...
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0answers
19 views

Is the projection of two isomorphic direct sum of modules isomorphic? [closed]

If A and B are R-modules, and if A x A is isomorphic to B x B, does this imply that A is isomorphic to B?
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0answers
23 views

Permutation modules and their vector space dimensions

I'm given a field $k$, a finite group $G$ and a set $S$ which $G$ acts on transitively. I'm then told to consider the permutation module $M = kS$. My first problem is understanding what the ...
1
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1answer
30 views

Krull-Schmidt theorem and internally cancellable modules?

According to this lecture notes (in Lemma2.1) the statement $Ae\simeq Ae^{\prime}\to A(1-e)\simeq A(1-e^{\prime})$ is true for finite dimensional algebras by using Krull-Schmidt theorem. Can anyone ...
0
votes
1answer
12 views

Statements about modules (generators and linearly independent sets)

I am trying to prove or disprove with a counterexample the following statements: (i) From every set of generators of a module $M$ one can extract a basis. (ii) Every linearly independent subset of a ...
0
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0answers
24 views

Presentation of a module by generators and relations

Let $R:=\mathbb C[T]$. Match the $R$-module with the presentation by generators and relations. $\bullet$$R$-modules: $M:=\mathbb C[T,T^{-1}]$ (Laurent Polynomials)$\qquad$$N:=\mathbb ...
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2answers
74 views

$k[x]/(x^n)$ module with finite free resolution is free

How to show a $k[x]/(x^n)$ module with finite free resolution is free? Suppose we have a exact sequence $k[x]/(x^n)^{\oplus n_1}\to k[x]/(x^n)^{\oplus n_{0}}\to M\to 0$, how do we get ...
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0answers
26 views

Are matroids really a generalization of independence in vector spaces?

The axioms of matroids are (Wikipedia): A finite matroid $M$ is a pair $(E,\mathcal{I})$, where $E$ is a finite set (called the ground set) and $\mathcal{I}$ is a family of subsets of $E$ (called ...
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0answers
10 views

Hopfian and Co-Hopfian Modules.

Can someone tell me why we want to study this class of modules? Let $M$ be a $R-$module. Wa say that : 1- $M$ is a Hopfian module, if every epimorphism of $M$ is a monomorphism. 2- $M$ is a ...
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2answers
37 views

An ideal which is not finitely generated

Let $K$ be a field and $A=K[x_1,x_2,x_3,...]$. Prove that the ideal $I:=\langle x_i: i \in \mathbb N\rangle$ is not finitely generated as $A$-module. I have no idea what can I do here, I mean, ...
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0answers
21 views

Semisimple algebras

In general, the tensor product of two algebras is not semisimple. Let $k$ be a ground field and let $A$ be a direct product of finite dimensional division $k$-algebras. Let $R=k^{n}$ where $n$ is ...
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0answers
27 views

Locally cyclic module exercise

An $A$-module $M$ is locally cyclic if every submodule of $M$ of finite type (finitely generated) is cyclic. (i) Show that every submodule of a locally cyclic module is locally cyclic. (ii) Prove ...
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1answer
11 views

Module of finite type has a minimal system of generators

I am trying to show the following statement: Every module of finite type, i.e. finitely generated, has a minimal system of generators (a minimal system of generators $\mathcal S$ is a system of ...
2
votes
1answer
110 views

Yoneda implies $\text{Hom}(X,Z)\cong \text{Hom(}Y,Z)\Rightarrow X\cong Y$??

I came across a result on the page Theorems implied by Yoneda's lemma? which said that Yoneda's Lemma implies the isomorphism of the title; namely, if we have $\text{Hom}(X,Z)\cong ...
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1answer
20 views

Under what conditions can I expect the restriction of scalars functor to preserve tensor products

Suppose I have the canonical injection $i:H\hookrightarrow G$. Evidently I can induce the map on modules which restricts scalars from $\mathbf{Z}[G]$ to $\mathbf{Z}[H]$; that is, ...
1
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1answer
64 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 $k$- 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, ...
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0answers
32 views

What is the tensor product of Z/10Z with Z/12Z? [duplicate]

I've just met tensor products of modules in part of my self-study and as a concrete exercise I've been asked to calculate $\mathbb{Z}/(10){\otimes}_{\mathbb{Z}} \mathbb{Z}/(12)$. Short of writing out ...
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1answer
24 views

Questions related to the concept of $k$-algebras

I am reading about modules and some days ago I've worked on some exercises related to $k$-algebras. The definition I've seen of $k$-algebra is that it is a field $k$ and a ring $A$ together with a ...
3
votes
1answer
36 views

Short exact sequence of modules

I am trying to show that if we have the following left splitting short exact sequence of $R-$modules: $0 \rightarrow M \stackrel{f} \rightarrow N \stackrel{g} \rightarrow S \rightarrow 0$ then there ...
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2answers
35 views

Why $\mathbb Z/ 2 \mathbb Z$ is not a free module?

I am reading some abstract algebra notes about free modules. It says that not all modules are free and the example to illustrate this is $\mathbb Z/ 2\mathbb Z$ (as a $\mathbb Z$-module) is not a free ...
5
votes
2answers
55 views

The rationals as a direct summand of the reals

The rationals $\mathbb{Q}$ are an abelian group under addition and thus can be viewed as a $\mathbb{Z}$-module. In particular they are an injective $\mathbb{Z}$-module. The wiki page on injective ...
0
votes
1answer
14 views

Morphism of $k$-algebras between abelian group of $n \times n$ matrices and $m \times m$ matrices

Problem Let $k$ be a field and $f:M_n(k) \to M_m(k)$ be a morphism of $k$-algebras ($n,m \in \mathbb N$). Prove that $n$ divides $m$. I have no idea what to do here, I thought that maybe I should ...
1
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1answer
32 views

Definition of Tensor Product of Modules

I am really struggling to understand several parts of the definition of tensor product given in my lecture notes: Definition of the tensor product *Denote by L the free A-module with a basis ...
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2answers
44 views

Is $R\otimes_R M\cong M$ when $R$ not necessarily commutative?

Suppose $R$ is not commutative. I am guessing in general that $R\otimes_R M\cong M$ fails to be true. However, are there any cases where this is still true? For instance group rings for a ...
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3answers
29 views

Definition of direct sum of modules?

I have just started studying modules, and am trying to figure out the definition of the direct sum of modules but I'm having trouble since different sources seem to give different definitions, for ...