Questions tagged [modules]
For questions about modules over rings, concerning either their properties in general or regarding specific cases.
8,707
questions
0
votes
0
answers
33
views
Extending epimorphisms from subspaces.
Suppose $U$ and $V$ are finite-dimensional vector spaces over some field $\mathbb{K}$, and $W$ is a subspace of $V$. We know that any homomorphism $f : U \to W$ can be extended to a homomorphism $f^* :...
- 77
2
votes
0
answers
49
views
A question about split exact sequence in rng category
For $R$-module category, we have the notion of split exact sequence which is given by the following.
For a fixed ring $R$, let $M,M^\prime,M^{\prime\prime}$ be modules over $R$.
We call an exact ...
- 170
0
votes
0
answers
30
views
Relationship between $M$ as an $A$-module and as a $B$-module, where $B=\operatorname{End}_A(M)$ [closed]
Let $A$ be a $K$-algebra, $M$ a left $A$-module and $B=\operatorname{End}_A(M)$. Show that:
a) If $M$ is a generator as a left $A$-module, then it is projective as a right $B$-module.
b) If $M$ is ...
3
votes
0
answers
23
views
Computing local cohomology over a noncommutative ring
Let $R$ be an $\mathbb{N}$-graded ring, and let $\Gamma : R\text{-gr.mod} \to R\text{-gr.mod}$ be the left exact functor defined by
$$
\Gamma(M) = \{ m \in M \mid R_{\geq n} m = 0 \text{ for some } n \...
- 63
1
vote
1
answer
38
views
Smith Normal forms in a polynomial ring
Let $M$ be a $\mathbb{C[x]}$ module with generators $m_1,m_2$ and relations : $$(x^2+ix)m_1+(x+i)m_2=0 \\ (-2x+2i)m_1+ (x^2+1)m_2=0 $$
Find integers $t,n_1,...,n_s \in \mathbb{N_0}$ and $\lambda_1,\...
- 576
1
vote
1
answer
26
views
Can one check if a map of infinite rank free modules is an isomorphism modulo the maximal ideal
If $(A,\mathfrak{m},k)$ is a commutative local ring, $M,N$ are two free modules, and you are given a map $M\to N$ such that the induced map $M\otimes k\to N\otimes k$ is an isomorphism, then was $M\to ...
- 169
0
votes
0
answers
33
views
How can I explain the intuition behind the First Isomorphism Theorem for modules or for rings to a lay person in mathematics
How can I explain the intuition behind the First Isomorphism Theorem for modules or for rings to a lay person in mathematics, like to a non-scientist. I want to use a very elementary example in life.
...
- 243
0
votes
0
answers
25
views
Distribution of a matrix product under modular arithmetic
The question I have is related to modular algebra. All algebraic statements I make henceforth are with respect to algebra of the module where arithmetic is over $\mathbb{Z}_q$ for general $q \geq 2$.
...
- 161
2
votes
1
answer
49
views
Let $M = \operatorname{cok}(\phi)$ where $\phi: \mathbb{Z}^3 \to \mathbb{Z}^3$. Express $M$ as a direct sum of cyclic modules.
Let $\phi: \mathbb{Z}^3 \to \mathbb{Z}^3$ be given by the matrix $A$ below. $$A = \begin{bmatrix}
2 & 2 & 4\\ 2 & 4 & 6 \\ 2 & 6 & 8
\end{bmatrix}$$ If $M = \operatorname{cok}(\...
- 1,906
0
votes
0
answers
25
views
Covariant functor preserves chain homotopy
Let $C_\bullet$ and $C'_\bullet$ be two chain complexes of $A$-modules, $A$ a $K$-algebra, and $f_\bullet,g_\bullet: C_\bullet \to C'_\bullet$ two homotopic morphisms. Show that, for any covariant ...
1
vote
1
answer
44
views
Short Exact Sequence Involving Ext Functor
Let $A$ be a $K$-algebra, and $0 \to M' \to M \to M'' \to 0 $ be an exact sequence of $A$-modules that splits. Show that
$$ 0 \to \operatorname{Ext}^n_A(X,M') \to \operatorname{Ext}^n_A(X,M) \to \...
3
votes
0
answers
89
views
Is submodule of a finitely generated free $\mathbb{Z}G$-module free when $G$ is free?
We know that a submodule of a free $R$-module is not necessarily free. But if $R$ is a PID, then every submodule of a free $R$-module is free.
Now let $G$ be a free group of finite rank $r>1$ and $...
- 1,691
1
vote
1
answer
47
views
$\operatorname{Hom}(L, \operatorname{Hom}(M,N)) \cong \operatorname{Hom}(M, \operatorname{Hom}(L,N))$
Let $L, M,N$ be three $A$-modules, $A$ a commutative ring. Show that $$\operatorname{Hom}_A(L, \operatorname{Hom}_A(M,N)) \cong \operatorname{Hom}_A(M, \operatorname{Hom}_A(L,N))$$
I know that, by ...
1
vote
0
answers
46
views
The proof of the Hilbert-Burch theorem
I am reading the proof of Theorem 1.4.17 (Hilbert-Burch theorem) in Cohen-Macaulay Rings by Bruns & Herzog.
My question
Let $R$ be a Noetherian ring, and $\varphi\colon R^n \to R^{n+1}$ an $R$-...
- 139
1
vote
0
answers
34
views
Modules over the trivial ring [duplicate]
I know that a module is the ring-theoretic analogue of a vector space. My question is, is every module over the trivial ring $\{0\}$ a singleton set? Or can there be a module over the trivial ring ...
- 13.6k
3
votes
0
answers
34
views
Bimodules over a categorical monad
In category theory a monad consists in an endofunctor $M\colon \mathcal{C}\rightarrow \mathcal{C}$ together with natural transformations $\mu \colon M \circ M \Rightarrow M$ and $\eta \colon Id_{\...
- 1,871
0
votes
0
answers
27
views
Grade and projdim of finite modules
I am reading Cohen-Macaulay Rings by Bruns & Herzog, and trying to prove the lemma below:
Lemma. Let $R$ be a Noetherian ring, and let $M$ be a finitely generated $R$-module. Then for all $\...
- 139
2
votes
2
answers
65
views
Demonstrate that $\mathbb{C}[\mathbb{Z}_{2} \times \mathbb{Z}_{3}] \simeq \mathbb{C}^{6}$.
This is a review problem that I'm solving. I have been told that
$ \mathbb{C}[\mathbb{Z}_{2} \times \mathbb{Z}_{3}] $ is a $ \mathbb{C} $ algebra so a vector space where the field is $\mathbb{C}$ and ...
- 354
0
votes
0
answers
28
views
Submodules of a complex vector space and invariants of an endomorphism
Let $V=\mathbb{C^n}$ be a finite dimensional vector space and $f\in End(V)$. Consider the $\mathbb{C[x]}$-module $\mathbb{C_f^n}$ which is obtained by giving $V$ the left-module structure as $p.v=p(f)(...
- 576
0
votes
0
answers
24
views
Chinese Remainder Theorem in Structure Theorem for f.g. Modules over a PID
I have been asked by my Algebra professor, to explicitly determine the Chinese Remainder Theorem in the proof of the Structure Theorem for f.g. Modules over a PID.
Here's what I know:
$\textbf{...
- 163
1
vote
0
answers
31
views
Understanding natural $R$-Algebra structure on some abelian groups.
Definition ($R$-Algebra): Let $R$ be a commutative ring with a ring homomorphism $\phi : R\to S$ such that $\phi(R)\subseteq Center(S)$, then the ring $S$ is a $R$-Algebra
Base Question:
$R$ is a ...
- 2,256
0
votes
0
answers
56
views
Are "modules over an additive group" a thing?
Googling around, the only notion of "module over a group $G$" I was able to find was here, where $G$ is a multiplicative group but acts on an additive group. But I'm curious about another ...
- 5,328
1
vote
2
answers
33
views
Is a finite ring map of finite presentation finitely presented?
Let $\phi: A \rightarrow B$ be a ring map (of commutative unitary rings).
Assume that $\phi$ is finite, i. e. $B$ is finitely generated as an $A$-module, and $\phi$ is of finite presentation as in ...
- 63
0
votes
0
answers
10
views
Decomposition of the ring of square matrices into indecomposable submodules [duplicate]
Let $R=K^{n\times n}$ be the ring of square matrices over field $K$, alongside left-multiplication we observe $R$ as a module over itself. Find a decomposition of $R$ as a direct sum of ...
- 576
1
vote
1
answer
61
views
Exact sequence in which $M/\operatorname{Ker}f$ and $\operatorname{Im}f$ appear [closed]
Let $R$ be a ring and $M,N$ be $R$-module.
Let $f:M→N$. I'm looking for an exact sequence in which $M,N,\operatorname{Im}f,M/\operatorname{Ker}f$ appear.
Do you have any good ideas?
Thank you for your ...
- 424
0
votes
1
answer
48
views
How do I show that if $R=\Bbb{Z}/6\Bbb{Z}$ and $M=\Bbb{Z}/3\Bbb{Z}$ then $M$ is a projective $R$-module?
Projective Module Let $R$ be a ring and $M$ an $R$-module. We say that $M$ is projective if $Hom(M,-)$ is exact.
Let $R=\Bbb{Z}/6\Bbb{Z}$ and take $M=\Bbb{Z}/3\Bbb{Z}$. Using the above definition I ...
- 1,235
-2
votes
0
answers
45
views
sets and logic question
Let S = {1, 2, 3, …, 19, 20}. Let ≡ be the equivalence relation on S defined by congruence modulo 7. a) Find the quotient set 푆 ≡. b) Find a system of equivalence class ...
2
votes
1
answer
82
views
If $M \otimes N$ is flat and $M$ is flat, then $N$ is flat?
So I was wondering if the product of two modules is flat then one of them being flat implies the other one is flat too. I think the answer is affirmative and I will try to present my reasoning here. I ...
- 461
0
votes
0
answers
40
views
Definition of 'Group action commutes with endomorphism'
Let $G$ be a group and $G$ acts on $R$-module $M$($R$ is a ring).
And $M$ has endomorphism $f$.
In these settings, what is the definition of $f$ commutes with $G$-action ?
I often hear this kind of ...
- 592
2
votes
1
answer
49
views
Motivation for looking at the product module
Let $M$ be an $A-$module, $A$ a commutative ring with identity. Then we have the following result that , $\frac{A}{I} \otimes M \simeq \frac{M}{IM}$ for any ideal $I$ in $A$.
This simply follows by ...
- 774
4
votes
1
answer
46
views
Is the Ext module $\mathrm{Ext}^i_R(R/I^n,-)$ annihilated by a power of $I$?
Let $R$ be a Noetherian ring and $I$ an ideal in $R$. Let $M$ be a finitely generated $R$-module.
Is, for all $i$ and $n$, the module $\mathrm{Ext}^i_R(R/I^n,M)$ killed by a power of $I$?
I've read ...
- 670
0
votes
0
answers
33
views
Tensor Algebra of Direct Sum
If $A$ is a $k$-algebra, and $(T(M), j_M)$ denotes the tensor algebra of an $A$-bimodule $M$, $j_M: M \to T(M)$ the associated homomorphism of $A$-bimodules, how can I prove that
$$T(M \oplus N)\cong ...
1
vote
0
answers
28
views
Elements of $\mathrm{Hom}_{\mathbb{Z}}(\prod_{i\geq 0}\mathbb{Z}, \mathbb{Z})$ vanish on almost all elements of "standard basis" $\mathbb{e}_{n}$
I've been struggling with the following exerciese:
Let $f\in\mathrm{Hom}_{\mathbb{Z}}(\prod_{i\geq 0}\mathbb{Z},
> \mathbb{Z})$, where $\prod_{i\geq 0}\mathbb{Z}$ denotes the
$\mathbb{Z}$-module ...
- 333
0
votes
0
answers
24
views
Could you give me an example of $R$module $M$ which is finitely generated but not finitely presented? [duplicate]
It is well known that finitely gpresented module is finitely generated.
But in general I heard finitely finitely module is not always presented represented.
Here, finitely represented $R$ module $M$ ...
- 592
-2
votes
0
answers
21
views
An example of $R$module $M$ which is finitely presented but not finitely generated [duplicate]
It is well known that finitely generated module is finitely presented.
But in general I heard finitely represented module is not always finitely represented.
Here, finitely represented $R$ module $M$ ...
- 592
3
votes
1
answer
61
views
Proposition 2.18 in Atiyah & MacDonald's Comm. Algebra
I understand the proof of the proposition, but I don't see how the functions in the tensored sequence will end up being $f\otimes 1, g\otimes 1$. In the first step of the proof, the functions change ...
- 63
1
vote
1
answer
31
views
Alternative proof for a sufficient condition for $x$ to be integral over a ring $A$.
I have a request for an alternative proof of a fact in ring theory. Given an extension of rings $R \supseteq A$, we say that $x\in R$ is integral over $A$ if there is a monic polynomial $p$ with ...
- 1,738
1
vote
1
answer
32
views
Tensor product of modules over Kronecker algebra
Let $\mathbf{k}$ be a field and $A=\begin{bmatrix}\mathbf{k}&0\\\mathbf{k}^2&\mathbf{k}\end{bmatrix}$ be the Kronecker algebra.
Let $M$ and $N$ be the left and right $A$-modules (respectively),...
- 588
0
votes
2
answers
65
views
How do I check if a module $M=\Bbb{Z}/6\Bbb{Z}$ is finitely presented?
Let $R=\Bbb{Z}$ and take $M=\Bbb{Z}/6\Bbb{Z}\in Mod_R$. I want to check if it is finitely presented.
I have the following definition:
If $M$ is an $R$ module generated by $(x_i)_{i\in I}$ then we ...
- 1,235
2
votes
0
answers
38
views
Homology of the diagonal sequence of 3x3 commutative diagram of modules
Suppose we have modules $M_{i,j}$ over a commutative ring $R$ (or members of some abelian category, like quasi-coherent sheaves of modules),
and suppose that we have a 3x3 commutative diagram, where ...
- 932
3
votes
1
answer
43
views
An example of abelian ring which is NOT square-free.
Recall that a ring $R$ is abelian if all idempotents are central.
Recall that a right module $M_R$ is called square-free if whenever $A$ and $B$ are submodules of $M$ with $A\cong B$ and $A\cap B=0$ ...
- 527
5
votes
1
answer
47
views
Why do we care about flat and projective modules?
I was reading the definitions of projective modules and flat modules and found myself a bit unenlightened (by all of their equivalent definitions). At least the Wikipedia articles for these classes of ...
- 435
1
vote
1
answer
24
views
Connection between kernels of linear maps of semimodules and injectivity
Let $S$ be a semiring (i.e. satisfies all the ring axioms besides existence of additive inverses) and $M, N$ semimodules over $S$ (same thing). For a linear map $\varphi : M \rightarrow N$, we can ...
- 402
0
votes
1
answer
18
views
Do we always have linear right inverses of surjective linear functions in non-free modules?
Let $R$ be any ring, $M$ an $R$-Module, not necessarily free.
Let $\varphi : M \rightarrow M$ be linear and surjective. By surjectivity, there exists a function $\varphi^*$ such that $\varphi \varphi^*...
- 402
2
votes
2
answers
77
views
Meaning of 'set of well-ordered sequences'
I'm trying to make sense of a construction of a module given in the following research paper: A New Construction of the Injective Hull, Fleischer, 1968.
On the second page, a module $F$ is constructed,...
- 2,550
2
votes
2
answers
82
views
Is the dual of a Noetherian module Noetherian?
Say $M$ is a Noetherian module over $R$. I wanna see how far I can get without the axiom of choice, so let me clarify that I mean "satisfies the Ascending Chain Condition" by Noetherian.
...
- 402
3
votes
0
answers
42
views
Proving $\mathbb{Z}_{(m,n)}$ is a submodule of $\mathbb{Z}_{n}$ and for every $[a]_{n}$ we get $[a]_{n}m=[0]$ where $(m,n)=\gcd(m,n)$ .
Proving $\mathbb{Z}_{(m,n)}$ is a submodule of $\mathbb{Z}_{n}$ and every $[a] \in \mathbb{Z}_{(m,n)}$ can be regarded as an element $[a]_{n}$ such that $[a]_{n}m=[0]$ where $(m,n)=\gcd(m,n)$ .
...
- 85
0
votes
0
answers
31
views
What are conditions on a ring $R$ for all finitely generated submodules of $R^n$ with torsion-free quotients to be free?
Let $R$ be a commutative integral ring, and let $M$ be a finitely generated sub-module of $R^n$ such that $R^n / M$ is torsion-free (that is, if $rx \in M$ then $x \in M$ for all $x \in R^n$ and $r \...
- 1
1
vote
0
answers
28
views
Admissibility necessary for a $(\mathfrak{g}, \mathfrak{k})$-module to be a direct sum of simple $\mathfrak{k}$-modules?
I am now looking over the book A. Borel, N. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, Volume 67 of Mathematical Surveys and Monographs, AMS. I would ...
- 370
1
vote
1
answer
38
views
Is every quotient module with identity cyclic?
If $R/\langle a \rangle$ is the quotient module over the commutative ring $R$ with identity 1, then,
$R/\langle a \rangle=\{r+\langle a \rangle: r \in R\}=\langle1+\langle a \rangle \rangle$
Is it ...
- 582