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

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4
votes
1answer
38 views

$M$ is a faithful $A$-module.

Let $A = M_2(\mathbb{C})$ and let $M = \mathbb{C}^2$ with its usual $A$-module structure. How do I show that $M$ is a faithful $A$-module? I understand that this amounts to showing that its ...
1
vote
3answers
46 views

Does there exist a module structure over $\mathbb{C}^2$ as a $\mathbb{C}[x]$-module?

Let $R = \mathbb{C}[x]$, $M = \mathbb{C}^2$. Does there exist a module structure over $M$ as an $R$-module or not?
2
votes
2answers
20 views

Module $M$ is infinite dimensional as a $\mathbb{C}$-vector space.

Let $M$ be a $\mathbb{C}[x]$-module. Since $\mathbb{C} \subset \mathbb{C}[x]$, we may consider $M$ as a $\mathbb{C}$-vector space. If $M$ is faithful, must $M$ be infinite dimensional as a ...
0
votes
1answer
28 views

Equivalence between category of $R$-modules and $S$-modules

Is it true that by equivalence between category of $R$-modules and $S$-modules we conclude that $R$ and $S$ are isomorphic?($R$ and $S$ are unital rings.)
5
votes
0answers
31 views

Exists an $R$-module homomorphism $\varphi: M \to \mathbb{Q}/\mathbb{Z}$ such that $\varphi(m) \neq 0$?

Let $M$ be an $R$-module with some torsion element $m$. Does there exist an $R$-module homomorphism $\varphi: M \to \mathbb{Q}/\mathbb{Z}$ such that $\varphi(m) \neq 0$?
0
votes
1answer
13 views

vector space constructed through a torsion module

Let $R$ be a principal ideal domain, $p \in R$ a prime element and $M$ a finitely generated $p$-torsion module of the form: $$ M = R/(p^{e_1}) \oplus \dots \oplus R/(p^{e_t}). $$ Let now be $_pM = ...
0
votes
2answers
19 views

Isomorphism of subquotient and quotient modules

Is it true that for every submodules $A\subset B$ of $C$ there is some submodule $L\subset C$ such that $C/L = B/A$?
0
votes
0answers
15 views

Split sequence and complement of a submodule

I have proved the equivalence of 1 and 2, but have no idea about 3.Could anyone offer some advice?
0
votes
0answers
12 views

What are all the $k$ dimensional unimodular subspaces of $\mathbb{Z}^n$?

I am trying to prove the following assertion - The set of subgroups of $(S^1)^n$ which are isomorphic by an element of $Aut((S^1)^n)$ to the standard copy $(S^1)^k$ is naturally parametrized by the ...
1
vote
1answer
19 views

Does the restriction of scalars functor preserve the quotient module construction?

I have two rings $R,\,S$ (not necessarily commutative - in the case I have in mind, just $R$ is commutative) such that $i:R\subset S$. Clearly, if we have a module $M$ over $S$, then we can restrict ...
0
votes
0answers
45 views

Abstract interpretation of isomorphism between tensor product with dual and hom

I'm interested in the following statement, coming from Remark 6.4.21 of Qing Liu's Algebraic Geometry and Arithmetic Curves: Let $\mathcal{F}, \mathcal{G}$ be quasi-coherent sheaves on a scheme ...
2
votes
2answers
47 views

Is the minimal number of generators of an ideal the rank of the ideal as a free $\mathbb Z$-module?

In an algebraic number theory course, my lecturer said that any ideal of $\mathcal O_K$, where $K$ is a quadratic number field, is generated by at most two elements. I am wondering why this is. When ...
2
votes
1answer
20 views

Describing the Kernel of an identity-preserving ring morphism from a ring $R$ to an Endomorphism ring of an additive Abelian Group.

I'm currently working through TS Blyth's book on Module Theory (Module Theory: An Approach to Linear Algebra). From Exercise 2.3: "Let $M$ be an Abelian Additive Group, and $R$ a unitary ring. Let ...
1
vote
2answers
29 views

Conductor of a ring

An easy (possibly trivial) question from Neukirch's Algebraic Number Theory, p.47 Let $A$ be a Dedekind domain, $K$ its fraction field, $L$ a finite separable extension of $K$ and $B$ the integral ...
0
votes
0answers
16 views

free submodules of free modules of a PID

Let $R$ be a principal ideal domain. Now I want to show that each submodule $N \subseteq M$ of a free $R$-module $M = R^{(I)}$ is also free. As a hint, it says I might consider the tripel $(J, ...
0
votes
2answers
36 views

Finitely generated modules, related to the Gaussian integers

Let $M = \mathbb{Z}[i] = \{a + bi|a, b \in \mathbb{Z}\}$ be the additive group of the Gaussian integers. Consider the $\mathbb{Z}$-submodules $N_1 = 2M$, $N_2 = (1 + i)M$. I now want to figure out ...
0
votes
0answers
30 views

The maps $\delta: V \to B \otimes V$ and $\delta': V \otimes V^* \to B$.

Let $B$ be a coalgebra and $V$ a vector space. Suppose that we have a coaction $\delta: V \to B \otimes V$. Is the map $\delta$ equivalent to a map $\delta': V \otimes V^* \to B$? Thank you very much. ...
1
vote
1answer
11 views

Can one extent to (left) modules on IBN ring the results of vector spaces?

We know that (left) modules "are not as nice as vector spaces" because lots of properties of these ones don't fit to the most general modules. But il we restrict to modules on IBN ring (i.e. modules ...
0
votes
0answers
24 views

properties of finitely generated torsion-modules and their submodules over a PID

Let $R$ be a principal ideal domain, $M$ a finitely generated $R$-torsion module, and $N \subseteq M$ a submodule. I want to show that there exist free R-modules $F, F', F''$ and module homomorphisms ...
0
votes
0answers
26 views

Does the dual behave well with base change?

Let $R$ be a ring, $M$ an $R$-module and $R \to S$ a ring homomorphism. I wondered under which conditions we have an isomorphism $$ Hom_R(M,R) \otimes_R S \xrightarrow{\sim} Hom_S(M \otimes_R S, S). ...
4
votes
0answers
54 views

If $\mathfrak{m}\otimes M\rightarrow A\otimes M$ is injective, what else has to be injective?

Let $A$ be a local (not necessarily noetherian) ring with maximal ideal $\mathfrak{m}$ and residue field $k$. Let $M$ be a finitely generated $A$-module such that $\mathfrak{m}\otimes_A M\rightarrow ...
1
vote
0answers
37 views

Need more insight on a formula

Following is a part of a programming contest problem. Given $C_{1},m,n,o,x,y,z,c,d,K,J$ are positive integers $ C_{i} = \left\{ \begin{array}{l l} (m*C_{i-1}^2 + n*C_{i-1} + o) \bmod J & ...
2
votes
1answer
49 views

When is composition in $A$-mod surjective?

Let $k$ be any field. Suppose that $A$ is a finite dimensional associative $k$-algebra. Let $A$-mod be the category of finite dimensional (over $k$) left $A$-modules. Then composition of ...
1
vote
1answer
12 views

How do I calculate the dimension of the localization of a module?

Apologies in advance if this is a vague question. So I am trying to calculate the $\mathbf{Z}$-rank of a finitely generated torsion free module $M$ (I won't go into the details of what this module $M$ ...
0
votes
1answer
39 views

Computing tensor products of $\mathbb{Z}$-modules.

I'd like to compute $\mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{Z}^{n}$, for some natural number $n$, and $\mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{Z}/n\mathbb{Z}$.
3
votes
2answers
36 views

Identity of tensor products over an algebra

I hope this question hasn't been asked before, but I wasn't sure what to search so as to check. Suppose $R$ is a commutative ring with 1 and that $A$, $B$ are unital $R$-algebras. Suppose further that ...
4
votes
0answers
48 views

When does the duality functor commute with the wedge power functor?

When working with modules over a fixed commutative ring, I know that $(M \otimes N)^* \cong M^* \otimes N^*$ provided either $M$ or $N$ is finitely generated projective. Does it follow that ...
1
vote
1answer
24 views

Factor Modules/Vector Spaces and its basis with canonical mappings

I'm having trouble with factor modules now. Well, specifically the following question from a past paper. Q. T=$R^2$ i.e. the real plane, and define $f:T$->$T$ with respect to the standard basis which ...
1
vote
1answer
12 views

Difference between an $R$-algebra being finitely generated and finite

So I have the two following definitions: An $R$-algebra $S$ is said to be finite over $R$ if it is finitely generated as an $R$-Module. An $R$-algebra $S$ is said to be finitely generated if $$S ...
0
votes
2answers
33 views

A short question about the direct limits and direct sum of commutative rings.

By viewing a ring as a $\mathbb{Z}$-module, it is possible to define the direct limit of rings following the same procedure for modules. Let's suppose to work with commutative rings with unity. It ...
2
votes
1answer
55 views

Under what conditions does a quotient module $A/B\cong C$ imply the direct sum $A\cong B\oplus C$?

Suppose we have some finitely generated $R$-modules $A,\,B,\,C$ such that $A/B\cong C$. Under what conditions is it necessarily true that $A\cong B\oplus C$? Clearly the converse to this is true, but ...
1
vote
0answers
43 views

What is the name of this special subset of module over commutative ring with unity

Let $R$ ba a commutative ring with unity and let $M$ be a $R$-module. For each $m,n\in M$ consider free spaces $\bar{m}$ and $\bar{n}$ generated respectively by $m$ and $n,$ i.e ...
1
vote
2answers
26 views

properties of transformations inbetween quotient modules

Let $R$ be a ring, $M$ and $R$-left module and $U, V$ submodules of $M$. I want to show that $$f: M/U \cap V \to M/U \oplus M/V, m + U \cap V \mapsto (m + U, m + V)$$ and $$g: M/U \oplus M/V \to ...
0
votes
1answer
29 views

existence of module homomorphisms, so that a diagram becomes commutative

Let $R$ be a ring, $f: R^k \to R^m$ aswell as $g: R^l \to R^n$ module homomorphisms and $M = coker(f)$, $N = coker(g)$. Let $p_M: R^m \to M$ and $p_N: R^n \to N$ be the natural projections. I now ...
1
vote
1answer
32 views

Existence of a Map

I just wanted to check to be sure I was correct as the more I stare at my proof the more I doubt myself. Suppose you have a commutative diagram of $R$-modules with exact rows: $$ ...
3
votes
1answer
19 views

Question about quotients of reducible elements and modules

First, I apologize if this is a stupid question. We started doing modules in my class a few days ago and I'm totally lost with the basics, I think. Suppose we have a ring $K$ and $M$ a $K$-module. ...
2
votes
0answers
18 views

Clarification about the definition of Direct limit of modules.

Let $A$ be a ring, $(M_i)_{i \in I}$ the family of modules that will bring the direct limit, and $f_{ij} : M_i \to M_j$ the transition maps. I have two definition of direct limit of $M_i$: Wikipedia ...
0
votes
3answers
70 views

Finitely Presented Modules Definition

I am a little bit confused with the definition of finitely presented modules. In Lang's Algebra he defines a module $M$ to be finitely presented if and only if there is a exact sequence $F'\to F\to M ...
2
votes
2answers
58 views

Basic rings (e.g. non commutative) $A$ such $A^n \simeq A^m$ and $n\neq m$

EDIT : precision and broadening of my question. Almost all is in the title : I am looking for various structures $A$ such $A^n$ and $A^m$ (products of $A$) are isomorphic (in the sense that it is ...
1
vote
1answer
23 views

Direct limit of modules: a property.

Suppose $A$ to be a ring and $M_i$ the indexed $A$-modules used to build the direct limit of modules $M \doteq \lim{M_i}$. Let $f_{ij}: M_i \to M_j$ the transition maps and $\phi : M_i \to M$ the ...
0
votes
1answer
18 views

Is an $R$-module $A$ a module over the image of a homomorphism $f:R\rightarrow{f(R)}?

Let $R$ and be a (unital) commutative ring with $A$ as an $R$-module. Now suppose $f\in{Hom(R,f(R))}$ I am wondering what requirements (if any exist) need to be placed on $f$ to ensure that $A$ is an ...
1
vote
0answers
31 views

Question about the K-Module $\cdot _\varphi : K[X]\times V \to V$

I managed to show several properties about the following mapping Let $K$ be a field, $V$ a finite dimensional $K$-Vectorspace and $\varphi \in \text{End}_K(V)$. $$ \cdot_\varphi : ...
0
votes
1answer
59 views

Localization commutes with direct sum.

Let $A$ be a commutative ring and $S \subseteq A$ a multiplicative subset. If $N$, $M$ are $A$-modules, is it true that $S^{-1} M \oplus S^{-1} N \simeq S^{-1} (M \bigoplus N)$? I need an ...
2
votes
2answers
78 views

Annihilators of elements of a finitely generated faithful module over a noetherian reduced ring

Lately I've been thinking to annihilator of modules and I've conjectured a proposition I can't prove, so I'll expose my claim. Let $A$ be a noetherian reduced (commutative) ring and let $M$ be a ...
6
votes
0answers
181 views

Elementary divisors theorem for Dedekind domains (Exercise in Lang's Algebra)

Exercise 13 (b) of Chapter III in Lang's Algebra is as follows. Let $M$ be a finitely generated projective module over the Dedekind ring $\mathfrak{o}$. Then there exists free modules $F$ and ...
2
votes
1answer
32 views

Product of characters in representation theory

Is it true that if $\phi_1$ and $\phi_2$ are characters then $\phi_1\phi_2$ is a character of a representation? I think this is true, say for instance if $R_1$ is a matrix representation with ...
1
vote
2answers
19 views

Ways to reduce polynomial modulus irreducible polynomial in F_2

Thank you all in advance for the help. I really appreciate your time in answering this question. I am trying to find a good summary of ways to reduce polynomial modules irreducible polynomial in ...
0
votes
1answer
47 views

A question about the definition of tensor product

Let $M$ and $N$ be modules over a ring $R$. Generally, the tensor product $M\otimes N$ is defined to be an abelian group with a balanced map $j:M\times N\to M\otimes N$ such that for any abelian group ...
0
votes
2answers
28 views

Is $R_1 \oplus 0$ a free $R_1 \oplus R_2$-module?

Suppose $R_1$ and $R_2$ are unital rings. Consider $R_1 \oplus \{0\}$ an $R_1 \oplus R_2$-module. Is this a free module? I am thinking it's not, since there are relations. How can I take this ...
1
vote
0answers
24 views

Infinitely many direct sum decompositions of $M$ into direct sum of irreducible $\mathbb{C}G$-modules?

I teaching myself character theory, but I don't understand a problem statement from Dummit and Foote, Exercise 18.3.4. Prove that if $N$ is any irreducible $\mathbb{C}G$-module, and $M=N\oplus N$, ...