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

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2
votes
2answers
190 views

Vector space as simple $K[x]$-module

I am trying to solve the problem: Let $V$ be a vector space and $T$ a linear transformation $T:V \to V$. Let $(V,T)$ be a $K[x]$-module. Show that $(V,T)$ is simple if and only if $V$ is finite ...
5
votes
4answers
2k views

Examples of faithfully flat modules

I'm studying some results about flatness and faithful flatness and I'd like to keep in my mind some examples about faithfully flat modules. In general, free modules are the typical examples. ...
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, ...
2
votes
1answer
57 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 ...
0
votes
0answers
27 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
2answers
40 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
1answer
75 views

Relation between two definitions of primary modules

Let $A$ be a commutative ring, $M$ be an $A$-module and $N \leq M$. There are two definitions of primary modules: 1) $M/N$ is coprimary (i.e., every zero divisor is nilpotent); 2) $\text{Ann}_A(N)$ ...
4
votes
0answers
61 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 ...
0
votes
0answers
32 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
12 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
28 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). ...
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
52 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 ...
0
votes
1answer
17 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 $M$ ...
6
votes
3answers
933 views

Learning to think categorically (localization of rings and modules)

I've been reading Ravi Vakil's notes on algebraic geometry notes and am having a hard time connecting the standard definition of the localization of a module to the definition in terms of universal ...
1
vote
1answer
45 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 $\mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{Z}/n\mathbb{Z}$.
0
votes
1answer
15 views

Length of quotients and relations between $\ell(\mathrm{coker}\varphi),\ell(R/\det\varphi)$

Let $R$ be domain(not necessary local) with maximal ideal $\mathfrak{p}$ and $d \in R, d \neq 0$. $(R/(d))_{\mathfrak{p}} = 0 \iff (d) \not\subset \mathfrak{p}$(?). And if $ (d) \subset \mathfrak{p}$ ...
3
votes
2answers
38 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 ...
1
vote
1answer
26 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 ...
6
votes
0answers
207 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 ...
1
vote
1answer
15 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 ...
1
vote
1answer
35 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: $$ ...
0
votes
2answers
48 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 ...
1
vote
0answers
58 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 ...
0
votes
1answer
31 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
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 ...
2
votes
1answer
106 views

Exercise from Rotman, direct limit of quotient modules

Suppose $$0 \to U \to V \to V/U \to 0$$ is an exact sequence of left $R$-modules. Let $\lbrace U_i, \alpha^{i}_j\rbrace$ be a direct sequence of submodules of $U$, where $$\alpha^{i}_j : U_i \to U_j ...
1
vote
1answer
28 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 ...
3
votes
1answer
22 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
26 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
1answer
77 views

Finitely Related Module is the direct sum of a Free Module and a Finitely Presented Module

Prove that any finitely related module may be expressed as the direct sum of a finitely presented module and a free module. Hint: If M is generated by X = X' U X'', where X' is the finite ...
2
votes
2answers
87 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 ...
2
votes
2answers
59 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 ...
3
votes
2answers
396 views

Example when the torsion of a module is not a submodule.

Can any one suggest me an example of a ring $R$ and an $R$-module $A$ s.t. torsion of module $A$ is not a sub-module? Torsion of module $A$, i.e. $\operatorname{Tor}(A)$, denotes all torsion ...
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 : ...
2
votes
1answer
43 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
25 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
48 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
1answer
36 views

Condition for module to not be free

Let $M$ be a module over a ring $R$ with $1$. Suppose that $rm=0$ for some nonzero $r \in R$ and $m \in M$. Can we conclude $M$ is not free? Here is my attempt. Suppose $M$ is free with basis ...
0
votes
1answer
17 views

Successive localizations of a module

I am trying to prove the following: Suppose that $p\subseteq q$ are prime ideals in $R$ and $M$ is an $R$-module. Then the the localization of the $R$-module $M_{q}$ at $p$ is $M_{p}$, i.e., ...
0
votes
2answers
31 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
25 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 $M=N\oplus N$, ...
3
votes
1answer
132 views

$M\simeq\ker(\beta)\oplus\frac{M}{\ker(\beta)}$

Let $A$ be a commutative ring. If $M,N$ are $A$-modules (not necessarily finitely generated), and $\beta:M\longrightarrow N$ is surjective, prove that ...
3
votes
1answer
246 views

Euler's theorem: [3]^2014^2014 mod 98

Calculate without a calculator: $$\left [ 3 \right ]^{2014^{2014}}\mod 98$$ I know I have to use Euler's Theorem. As a hint it says I might need to use the Chinese Remainder theorem too. I know ...
1
vote
1answer
41 views

Does $\text{End}_K(K^n) \cong \text{Mat}(n\times n, K)$?

Let $K$ be a field, $K^n$ a vector space over $K$. Is the following true? $\text{End}_K(K^n) \cong \text{Mat}(n\times n, K)$ Does this change if $K$ is a ring, and $K^n$ a module over $K$?
1
vote
1answer
37 views

Why is $\text{End}_K(K) = K$, or, more generally, $\text{End}_D(D) = D^{\text{op}}$?

Let $K$ be a field and a vector space over itself, $D$ a division ring with a modul over itself, with its opposite ring $D^{\text{op}}$. Why is then $\text{End}_K(K) = K$? Or more general, why is ...
2
votes
1answer
23 views

Character of action of permutations on a subset

I have a problem involving characters of a certain $\mathbb C$$G$-module, where $G = S_n$. The module is the vector-space $V$ with basis $\{v(I) | I\subset\{1,...,n\},|I| = k\}$, where $k\le n$. With ...
0
votes
1answer
14 views

Proof that $f:\mathbb CG \to S$ given by $f(a)=a\cdot s$ is surjective

I am going through a proof of the following result from my lecture notes: Suppose $\mathbb C G \cong \bigoplus_{i=1}^nU_i$ where the $U_i$ are simple $\mathbb CG $ modules. Then any simple $\mathbb ...
3
votes
1answer
38 views

Depth comparison on short exact sequences

Let $R$ be a Noetherian ring and $M,N,U$ be $R$-modules. We have a short exact sequence $$0 \longrightarrow U \longrightarrow M \longrightarrow N \longrightarrow 0.$$ We know that ...