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

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2
votes
1answer
24 views

Inducing a module homomorphism

Suppose that I have a homomorphism of $A$-modules $M \xrightarrow{f} N$ ($A$ commutative with 1), a ring homomorphism $A \xrightarrow{g} A'$, and two $A'$-modules $M'$ and $N'$. Suppose further that ...
1
vote
1answer
46 views

Limit of inverse system where morphisms are non-zero products

I have a rather complicated inverse limit which I am trying to compute. I will try to distil the question to its barest form, but if more details are necessary I can supply them. Let $R$ be an ...
1
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0answers
54 views

Resolution of module over polynomial ring

The problem is: Let $F$ be a field, and let $R = F[x_1, \ldots, x_r]$, the polynomial ring over $F$. Consider the $R$-module $M = R/(x_1, \ldots, x_r) \cong F$. Find a resolution of $M$ by free ...
2
votes
1answer
62 views

Structure of a module and finite fields

I am trying the following problem which appeared on an exam for a grad course: Determine the structure of finite field $F$ as a module over $F_p[x]$ when $xv=Lv$. Here $L(z)=z^p$ and ...
0
votes
1answer
57 views

$End(R[x])$ and $Hom(R[x],R)$

Is $End(R[x])$ (the set of all R-module homomorphism from $R[x]$ to $R[x]$) isomorphic to $R[x]$, for some ring $R$? Also, Is $Hom(R[x],R)$ (the set of all $R$-module homomorphism from $R[x]$ to $R$) ...
6
votes
1answer
69 views

Describe all the quotient groups of $\mathbb{Z} \oplus \mathbb{Z}$

I'm thinking about a problem in algebraic topology of how to determine all the $k$-fold covers of $\mathbb{T}^2$, where $k$ is a certain integer. In thinking about the problem, I came upon the ...
1
vote
1answer
29 views

The existence of a projective resolution of M from finite rank free modules

Any R-module admits a projective resolution. When the ring is Noetherian, can we show the existence of a projective resolution of any finitely generated R-module by finite rank free R-modules? In ...
2
votes
0answers
41 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 ...
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0answers
30 views

Classify all possible $R$-module structures on a vector space

Let $V$ be an $n$-dimensional complex vector space. In particular, it is an Abelian group. Let $R$ be a (commutative, unitary) $\mathbb C$-algebra. Problem. I would like to parameterize all ...
2
votes
1answer
38 views

When is a ring homomorphism also a R-module homomorphism?

Given a ring $R$(with or without $1$,may or may not be commutative), $R$ is also an $R-$module. Let $\phi_{ring}\colon R \to R $ be a ring homomorphism and $\phi_{module} \colon R \to R$ be a left ...
3
votes
1answer
103 views

Do these facts about $\mathbb{Q}/\mathbb{Z}$ generalize to more general cases?

We know that $\mathbb{Q}/\mathbb{Z} $ is injective. Not only that $0 \to A \to B \to C \to 0$ is exact iff $0 \to Hom(C,\mathbb{Q}/\mathbb{Z}) \to Hom(B,\mathbb{Q}/\mathbb{Z})\to ...
1
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2answers
56 views

Does tensoring with a module preserve this injection even if the module is not flat?

Suppose $R$ is a domain and let $Q$ be its field of fractions. Then $ 0 \to R \to Q $ is exact. Now suppose that $M$ is a torsion free $R$ module. Is it necessary that $ 0 \to M \to Q \otimes M $ ...
3
votes
2answers
65 views

Question about the set of all $\mathfrak p\in\operatorname{Spec}R$ such that $M_{\mathfrak p}=0$.

Is the following subset of $\operatorname{Spec}(R)$ always open? $R$ is a commutative, unitary ring and $M$ is an $R$-module.$$\{\mathfrak p \in\operatorname{Spec}(R) \colon M_{\mathfrak p} = 0\}$$
0
votes
0answers
36 views

Simple module over matrix rings

I'm trying to prove that if $M$ is a simple module over $M_n(D)$ where $D$ is a division algebra, then $M\cong D^n$. I know that if $M$ is a simple module over $R$ then it is isomorphic to ...
1
vote
1answer
29 views

Is there an analogue of Bass-Papp theorem for Projective modules?

The Bass-Papp theorem for injective modules states that If $R$ is a commutative ring such that every direct sum of injective $R$ modules is injective then $R$ is Noetherian. Is there an ...
-1
votes
1answer
39 views

e * d = 1 mod phi(n); How do I find d? [closed]

given suppose e = 5, phi(n) = 96. How do i find the value of d? How do I solve this problem?
0
votes
1answer
28 views

Semi-simple and characteristic

Let $C_3$ be the cyclic group of order 3, then let $K$ be a field, let $X=span_K(1+c+c^2)$, show that $X$ has a complement if and only if characteristic of $k$ is not equal to $3$.
2
votes
1answer
75 views

Exercise 2.26 Atiyah-Macdonald, flatness

I'm stuck on this exercise. $A$ is a commutative ring with unit. $N$ is an $A$-module. Then $N$ is flat $\Longleftrightarrow $ $\text{Tor}_{1}(A/a, N ) = 0 $ for every finitely generated ideal $a$ of ...
2
votes
1answer
74 views

Endomorphisms of direct sum and division ring

How to prove that $$\operatorname{End}_R(V^{ \oplus n }) \cong M_n(D),$$ where $V$ is a simple left $R$-module and $D=\operatorname{End}_R(V)$. This is part of the proof finally leads to prove ...
3
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0answers
64 views

Approximating modules over complete local rings

Let $A$ be a complete local noetherian ring with maximal ideal $\mathfrak{m}$. Is the canonical functor $$\mathsf{Mod}(A) \to \varprojlim_n ~ \mathsf{Mod}(A/\mathfrak{m}^n),~ M \mapsto ...
0
votes
1answer
53 views

Short Exact Sequences

Let $M \ge N \ge P$ be R-modules. Prove that there exist natural (not depending on choices) R-homomorphisms $N/P \to M/P$ and $M/P \to M/N$ for which the sequence $0 \to N/P \to M/P \to M/N \to 0$ is ...
0
votes
0answers
26 views

Splitting of Abelian Groups in Graded Modules

Suppose I have a graded module $M=\oplus M_i$ over a graded ring $R=\oplus R_i$. Assume $M_i\cong A \oplus B$, where A and B are abelian groups. Since $M$ is a graded ring, we know that $R_k(A\oplus ...
3
votes
1answer
51 views

Classify abelian groups $A$ which are irreducible $End(A)$-modules

Classify abelian groups $A$ which are irreducible $End(A)$-modules. I think i did it for finite abelian group $A$ . A finite abelian group $A$ is irreducible iff order of $A$ a is power of prime. ...
0
votes
2answers
52 views

Finitely generated quotient field

If $K$, the quotient field of a commutative integral domain $R$, is finitely generated as an $R$-module, is $R$ necessarily a field? Thanks for any help.
0
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0answers
38 views

Representation theory& module

$V$ is a left $R$ module, how do you understand the ring homomorphism $$\rho_{V}:R \to End_Z(V)$$ I know that it is like a group acting on sets, but it is very easy to understand like a group $S_n$ ...
0
votes
1answer
37 views

Definition of modules

The definition of modules confuses me: $R$ is a ring, then a left $R$ module is an abelian group $V$ together with a multiplication map $$R \times V \to V, (r,v) \to rv$$ satisfies some natural ...
0
votes
1answer
66 views

give an example of Hom(M,N) and Hom(N,M) are not isomorphic as R modules

give an example of a commutative unitary ring R and two finitely generated R modules M and N such that Hom(M,N) and Hom(N,M) are not isomorphic as R modules.
0
votes
1answer
32 views

Rings act on modules

About the definition of modules: $R$ is a ring, then a left $R$ module is an abelian group $V$ together with a multiplication map $$R \times V \to V, (r,v) \to rv$$ satisfies some natural axioms. I ...
2
votes
2answers
63 views

$R$-modules are also $\mathbb Z$-modules

Simple question: Why all $R$-modules are also $\mathbb Z$-modules and $$End_R(V) \subseteq End_{\mathbb Z}(V)$$ is a subring?
0
votes
2answers
55 views

example of tensor product of three finitely generated R-modules that equal 0

Can you give an example of a commutative unitary ring $R$ and three finitely generated $R$-modules $L$, $M$ and $N$ such that $L\otimes M\ne 0$, $M\otimes N\ne 0$, and $L\otimes N\ne 0$, but $L\otimes ...
0
votes
0answers
26 views

Basis of a module

I know that not all modules have bases, and those that do are called free modules. I know that all vector spaces have bases, and that a module $M$ over $R$ becomes a vector space if $R$ is a division ...
0
votes
2answers
68 views

Relationship between R and a submodule of a R-module

Consider a commutative ring with unity, call it $\mathbf{R}$. Suppose that $\mathbf{S}$ is a finitely generated $\mathbf{R}$-module and has a submodule $\mathbf{T}$. Then I was wondering if it holds ...
1
vote
1answer
48 views

Finitely generated torsion module over a PID

I have a question about modules over PID's from Lang's Algebra p. 152. Assume $E$ is a f.g. non-zero torsion module over a pid $R$. Lang shows that $E\cong R/(q_i)\oplus\cdot \cdot \cdot \oplus ...
0
votes
1answer
37 views

The sign representation of the Symmetric Group

I am currently trying to learn some of the basics of Representation Theory through Sagan's The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions. On page 11, after ...
1
vote
1answer
42 views

Modules over commutative rings

I'm trying to get my head around modules, and there's a problem that's bothering me regarding scalar multiplication from the left vs from the right. In many books/articles I've read, the author may ...
3
votes
0answers
69 views

exact sequence and modules proposition.

I have problems to prove the following proposition: Let's consider $$0 \rightarrow L \stackrel{\alpha}{\rightarrow} M \stackrel{\beta}{\rightarrow} N \rightarrow 0$$ an exact sequence of modules and ...
0
votes
1answer
25 views

How to show a Hom group is isomorphic to some modulo group?

Let $\mathbb{Z}_{n}[m]=\{a\in{\mathbb{Z}_{n}|ma=0\}}$, where $m$ is a positive integer. Then how do we establish the following? $$ \mathbb{Z}_{n}[m]\cong{\mathbb{Z}_{\gcd(m,n)}} $$
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0answers
35 views

Prove that $M \otimes N$ is isomorphic to $N \otimes M$.

Let $M,N$ be left $R$-modules with the standard $R$-module structures (making them bi-modules). The proof that $M \otimes N$ is isomorphic to $N \otimes M$ over a commutative ring $R$ in Dummit and ...
1
vote
3answers
52 views

How to show two groups are isomorphic?

For any abelian group $A$ and a positive integer $m$ prove that $$ Hom(\mathbb{Z}_{m},A)\cong{A[m]=\{a\in{A}|ma=0\}}. $$ I am not sure how to start to do this problem. Thanks in advance for your ...
2
votes
0answers
55 views

Why $\mathbb{Q}$ is not a projective $\mathbb{Z}$-module?

From the fact that $\operatorname{Hom}_{\mathbb{Z}}(\mathbb{Q},\mathbb{Z})=0$, how do we conclude that $\mathbb{Q}$ is not a projective $\mathbb{Z}$-module?
0
votes
2answers
35 views

Extension of R-modules.

Let $$ 0\longrightarrow{A}\longrightarrow{B}\longrightarrow{C}\longrightarrow{0} $$ be a short exact sequence of $R$-modules $A,B,C$ then why do we call $B$ is an extension of $C$ by $A$?
2
votes
0answers
26 views

The embedding $L^2(\Gamma(N)\backslash\text{SL}_2(\mathbb{R})) \hookrightarrow L^2(\text{SL}_2(\mathbb{Q})\backslash \text{SL}_2(\mathbb{A})))$?

Let $\mathbb{A}$ be the ring of adeles. Let $N$ be a natural number. Let $\Gamma(N)=\ker(\text{SL}_2(\mathbb{Z})\rightarrow\text{SL}_2(\mathbb{Z}/N\mathbb{Z})))$. I saw the following embedding of ...
0
votes
1answer
71 views

Prove a module is Noetherian

Let $R$ be a commutative Noetherian ring. Prove that an $R$ module $M$ is noetherian iff $M$ is finitely generated. One way is obvious. The other I have to prove every submodule of $M$ is finitely ...
2
votes
1answer
43 views

Projectiveness and Dedekind domains

Let $A$ be a commutative ring with unity and $M$ an $A$-module. Show that $M$ is flat if and only if $M_\mathfrak{m}$ is a flat $A_\mathfrak{m}$-module for all maximal $\mathfrak{m}\subseteq A$. ...
1
vote
2answers
69 views

$\text{V}$ is semisimple as a $k[X]$-module if and only if $A\in\operatorname{End}_k(\text{V})$ is diagonalizable over the algebraic closure of $k$

Suppose $\text{V}$ is a finite-dimensional $k$-vector space and let $A\in \operatorname{End}_k\left(\text{V}\right)$. Regard $\text{V}$ as a $k[X]$-module via $f(X)v=f(A)v$ for $f(X)\in k[X]$, ...
0
votes
1answer
20 views

The module $M/M_n$ is of finite length

I have this doubt from a proposition in Dimension theory from Atiyah Macdonald. The proposition is as follows. Let $A$ be a noetherian local ring, $\mathcal{m}$ be its maximal ideal, $\mathcal{q}$ an ...
3
votes
1answer
70 views

Kernel of a morphism of regular rings.

Let $k$ be a field and $f: A \rightarrow B$ be a surjective ring morphism between smooth Noetherian $k$-algebras. By smooth I mean that the module of Kahler Differentials $\Omega_{A|k}$ is a ...
2
votes
0answers
59 views

Question about definition of some classes of bimodules.

Suppose that we have a ring $R$ and a $R$-$R$ bimodule $M$ such that: For every $r\in R$ and $m\in M$ there exists $r'\in R$ such that $m\cdot r=r'\bullet m.$ Examples of this bimodules can be seen ...
3
votes
2answers
120 views

Every vector space has a basis using minimal spanning set.

We have seen the argument for proving the above statment using Zorn's Lemma by asserting the existence of a maximal linearly independent set which serves a basis. In finite dimensional vector space, a ...
4
votes
1answer
30 views

If $M$ is Noetherian, then $R/\text{Ann}(M)$ is Noetherian, where $M$ is $R$-module

Let $M$ be a $R$ -module and $\text{Ann}(M)=\{r \in R: rm =0 , \forall m \in M\} .$ Suppose $M$ is Noetherian, could anyone advise me on how to prove $R/\text{Ann}(M)$ is also Noetherian? Hints ...