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

learn more… | top users | synonyms

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 $M$ ...
1
vote
1answer
41 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
49 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
13 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
37 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
56 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
45 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
22 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
72 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
26 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
62 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
81 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
193 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
34 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
20 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
2answers
29 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 $M=N\oplus N$, ...
0
votes
1answer
33 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
15 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., ...
1
vote
1answer
40 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 $K$?
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 ...
2
votes
1answer
22 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 ...
1
vote
1answer
22 views

Castelnuovo-Mumford regularity of a shifted module

Let $R$ be a graded ring and $M$ be a graded $R$-module. Regularity definition $\operatorname{reg}(M)=\max\{j-i\mid \beta _{i,j}(M) \not=0\} $. What is the relation between ...
0
votes
1answer
17 views

every projective module has a free complement.

I have to prove that every projective module has a free complement. Now Rotman ask it to first do for $R=Z/6Z$ and $P=Z/2Z$, We know $Z/6Z \cong Z/2Z \oplus Z/3Z$. Now $Z/2Z$ is projective as it a ...
3
votes
1answer
34 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 ...
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 ...
1
vote
1answer
14 views

Is the image of a ring homomorphism an ideal?

Let $\phi : R \rightarrow S$ be a ring homomorphism, then is $im(\phi)$ an ideal in $S$? I ask this because I am studying about modules and in that we say that for a given $R$-module homomorphism the ...
1
vote
1answer
13 views

Relation between splitness of short exact sequences and free modules

I just proved a result , that, a short exact sequence with a free module at the third position is split exact. i.e. The short exact sequence, $0\to \mathbf M' \to \mathbf M \to \mathbf M'' \to 0 $ ...
1
vote
1answer
13 views

Question on the argument proving primary decomposition theorem

Lang - Algebra p.150, Lemma 7.6 Let $E$ be a torsion module of exponent $p^r(r\geq 1)$ for some prime element $p$. Let $x_1\in E$ be an element of period $p^r$. Let $\bar E = E/(x_1)$. Let ...
2
votes
1answer
96 views

An exercise using Nakayama's lemma.

Let $A$ be a ring and $\mathfrak a \subseteq A$ an ideal. Let $N \to M$ be a homomorphism of $A$-modules such that the induced homomorphism $N/\mathfrak a N \to M/\mathfrak a M$ is surjective. If $M$ ...
2
votes
1answer
37 views

Prove that $I_k \otimes_k \Omega \rightarrow I$ is injective

Let $\Omega$ be an algebraically closed field, $k$ a subfield of $\Omega$, $I$ an ideal of $\Omega[X_1, ... , X_n]$, and $I_k = I \cap k[X_1, ... , X_n]$. Then $I_k$ is an ideal of $k[X_1, ... , ...
2
votes
1answer
18 views

Presented matrix and number ring.

Let $V$ be the module generated by the column matrix $A= (2, 1+ \sqrt{-5})^T$. Prove that the residue of $A$ in $\mathbb{Z}[\sqrt{-5}]/ \mathfrak{P}$ has rank 1 for every prime ideal $\mathfrak{P}$ of ...
2
votes
2answers
54 views

Existence of projection in proof of Maschke's theorem

In a proof of Maschke's theorem, my lecturer writes "If $N$ is a $\mathbb C G$-module and $N\leq M $ is a submodule, let $\pi: M \to M$ be a projection (i.e. a map with $\pi^2=\pi$) with image $N$." I ...
1
vote
2answers
61 views

An exercise on tensor product over an integral domain.

This post is the natural conclusion of another one (An exercise on tensor product over a local integral domain.). Let $M$ be a finite module over an integral domain $A$. Let $Q$ be its fraction ...
1
vote
0answers
68 views

Let $R$ be a domain. Then $\operatorname{Tor}_n^R(A,B)$ is a torsion module

I have some problem to understanding the proof of this problem. This theorem is on page $414$ introduction to homological algebra Rotman. The theorem says: If $R$ is a domain, then ...
3
votes
1answer
220 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 ...
2
votes
2answers
53 views

How to show fraction field is flat (without localization)

Here I asked that if one can prove the field of fraction of a domain is flat. The answers used localization, which I am not familiar with. Can anyone prove it without using localization?
1
vote
2answers
46 views

Why field of fractions is flat?

I want to show this lemma: Let $R$ be a domain. If $A$ is a torsion $R$-module, then $\operatorname{Tor}_1^R (K,A)\cong A$ where $\operatorname{Frac}(R)=Q$ and $K=Q/R$. When I was reading ...
3
votes
1answer
57 views

Injection and surjection over free modules.

Let $A$ be a commutative ring and $M$ an $A$-module. Suppose to have both an injection $A^s \to M$ and a surjection $A^s \to M$ of module homomorphisms. Show that $M \simeq A^s$. This point is ...