Tagged Questions

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

learn more… | top users | synonyms

4
votes
2answers
39 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 ...
2
votes
2answers
30 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 ...
-1
votes
1answer
13 views

$\rm{End}_\mathbb Z(\mathbb Z_n)$ [on hold]

Let $\mathbb Z_n$ be $\mathbb Z$-module. I am looking for $\rm{End}$$_\mathbb Z(\mathbb Z_n)$ where by $\rm{End}$$_\mathbb Z$ I mean the ring of all $\mathbb Z$-module endomorphisms.
1
vote
1answer
25 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 ...
1
vote
0answers
6 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
votes
1answer
11 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 ...
0
votes
0answers
17 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?
1
vote
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 ...
1
vote
0answers
56 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
votes
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
vote
1answer
18 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 ...
1
vote
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
vote
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 ...
0
votes
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 ...
0
votes
1answer
42 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
votes
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
votes
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, ...
0
votes
1answer
52 views

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

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
votes
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 ...
0
votes
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
votes
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
vote
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 ...
1
vote
1answer
31 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 ...
0
votes
0answers
52 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 ...
0
votes
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 $A\otimes B\to A\times B$?
-3
votes
0answers
113 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 ...
1
vote
1answer
22 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 ...
3
votes
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
votes
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 $f(x) = g(x)$: ...
0
votes
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 $S$ ...
1
vote
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
vote
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 ...
1
vote
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
vote
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
votes
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 ...
1
vote
2answers
76 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 ...
0
votes
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 ...
0
votes
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 ...
1
vote
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, ...
1
vote
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 ...
1
vote
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
111 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 ...
1
vote
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
vote
1answer
65 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, ...
1
vote
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 ...
0
votes
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 ...
1
vote
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 ...