Questions tagged [modules]
For questions about modules over rings, concerning either their properties in general or regarding specific cases.
9,317
questions
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Subcategory determined by composition series
Suppose $A$ is an artin algebra and take the category $\operatorname{mod}A$ of finitely generated $A$-modules. Consider the following construction. Let $M$ be an $A$-module. Since $A$ is artinian, $M$ ...
- 329
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1
answer
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Irreducible module over upper triangular matrices
Let $R$ be a ring and $M$ be an irreducible $R$ module. I want to create an irreducible $A$-module, where
$$A = \begin{pmatrix}R & R\\0 & R\end{pmatrix}.$$
I defined $\bar{M}$ as
$$\bar{M} = \{...
- 394
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0
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33
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Facts about Weyl algebra
I am trying to prove a few things about the first Weyl algebra, $W = k[x,y]/(xy-yx-1)$ over an algebraically closed field $k$ with $char(k)=p>0$. In particular, I am interested in nilpotent ...
- 725
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Epimorphism and Surjective Homomorphsim [duplicate]
Here is a question I came across recently:
If a morphsim in Grp (category of groups) is an epimorphism, then it is a surjective group homomorphsim.
I believe it boils down to show that for any group $...
- 75
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answers
47
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Finitely generated ideals of integer valued polynomials
Let $\operatorname{Int}(\mathbb{Z}):= \{f(x) \in \mathbb{Q}[x] \mid f(\mathbb{Z}) \subseteq \mathbb{Z}\}$. Any element of the integer valued polynomials can be written as a $\mathbb{Z}$ linear ...
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11
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Chain map sequence $0 \to H(C) \to C/B(C) \xrightarrow{d} Z(C)[-1] \to H(C)[-1] \to 0$ is exact for a cochain complex of $R$-modules?
Specifically, Weibel Page 10 Exercise 1.2.7b.
I think I proved Exercise 1.2.7a or that there exists an SES:
$0 \to Z(C) \to C \xrightarrow{d} B(C)[-1] \to 0$
given a complex of $R$-modules $C^{\cdot}$ ...
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Example about endomorphism ring of an indecomposable object, which have non-trivial idempotent element in an additive category.
Recently I read some about Krull-Remak-Schmidt category.
If $A$ is an additive category in which every idempotent splits, every object is the biproduct of finitely many indecomposable objects and the ...
- 348
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25
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notation for creating formal power series and $R-module$
The following is taken from "Module Theory an approach to linear algebra" by T.S Blyth
$\color{Green}{Background:}$
$\textbf{(1) Example}$
Let $R$ be a unitary ring and let $R^N$ denote the ...
- 2,715
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44
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Example 2.2 in Rotman's Homological Algebra Text
1. Example 2.2 (excerpt)
2. Questions
If $x,y \in G$ are distinct, then in $kG$ we have $xy = \delta_x \delta_y = \delta_{x,y} = 0$. I think this is somehow wrong. The example says that $1 \in G$ is ...
- 585
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2
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$M$ being the direct sum of submodules $(M_i)_{i \in I}$ is equivalent to a certain map (between $\operatorname{Hom}$-sets) being an isomorphism.
I am currently trying to prove a remark in Bosch: Algebraic Geometry and Commutative Algebra (chapter 1.4):
Consider a family $(M_i)_{i \in I}$ of submodules in $M$. Then the inclusion maps $\iota_i \...
- 416
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answers
28
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Trouble understanding torsion
I'm reading this paper. In definition 6.1.2 it is mentioned that $H_n(G;A) \cong \text{Tor}_n^{\mathbb{Z}G}(\mathbb{Z}, A)$ (for all $n=0, 1, 2, ...$). What exactly is meant by the right-hand side? I'...
- 1,074
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Counterexample to $\hat{\mathfrak{a}}\hat{M}=\widehat{\mathfrak{a}M}$ when the base ring is not Noetherian or the module $M$ is not finitely-generated
$\def\fra{\mathfrak{a}}$Here it is proven that for $A$ a Noetherian ring, $\fra\subset A$ an ideal and $M$ a finitely-generated $A$-module and if we take $\fra$-adic completions, then $\hat{\mathfrak{...
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Is this solution to show that the Ideal given by the kernel of$ f \in R \mapsto f(0,0) \in \mathbb C[X,Y]$ is not finitely generated correct?
Let $R \subseteq \mathbb C[X,Y]$ be the subring of all polynomials $f \in \mathbb C[X,Y]$ that can be written as $f = g(X)+X ·h(X,Y)$
a. Let $I \subset R$ be the kernel of the evaluation map $R \...
- 2,696
1
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1
answer
72
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Jordan-Holder theorem for group algebras
I'm currently studying the Jordan-Holder theorem for modules and representations of associative algebras over fields. I was wondering if there is a way to prove the Jordan-Holder theorem for finite ...
0
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1
answer
68
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The Ideal given by the kernel of map $ f \in R \mapsto f(0,0) \in \mathbb C[X,Y]$ is not finitely generated for this particular polynomial subring
In the following problem, $R$-module means left $R$-module and $R$ is a ring.
I have already proven these facts that may or not be needed:
-Show that an $R$-module $ M$ is finitely generated if and ...
- 2,696
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1
answer
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Let $p$ be an odd prime number, and let $m \ge 0$ and $N \ge 1$ be integers. Let $\Lambda$ be a free $\mathbb{Z}/p^N\mathbb{Z}$-module of rank $2m+1$,
$$
\text{Let } p \text{ be an odd prime number, and let } m \ge 0 \text{ and } N \ge 1 \text{ be integers. Let } \Lambda \text{ be a free } \mathbb{Z}/p^N\mathbb{Z} \text{-module of rank } 2m+1, \text{...
- 386
1
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1
answer
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Is being finitely generated a local property
Searching on this site and others leads to lots of dicussion about localisation at multiplicatively closed subsets of the form $\{f_i^j\}_{j=1}^\infty$ where $\{f_i\}_{i=1}^n$ generate the whole ring ...
- 733
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Not sure how to show a module homomorphism to a finitely generated module is an isomorphism.. [duplicate]
I have the question: Let $M$, $N$ be modules over a ring $R$ with homomorphisms $f, g : N \longrightarrow M$
such that $f$ is surjective and $g$ is injective. Show that:
(1) $f$ is an isomorphism if $...
2
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0
answers
39
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Proving a lemma about an extending $R$ module with no $M$-singular direct summands in "Extending modules" (1994)
I'm reading this book:
Dung, N. V., Van Huynh, D., Smith, P. F., & Wisbauer, R. (1994). Extending modules (Vol. 313). CRC Press.
In the proof of Lemma 11.1, there is a part that I can not ...
- 135
2
votes
1
answer
42
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Alternative proof for Structure theorem for finitely generated modules over a principal ideal domain
I'm thinking of an alternative proof for Structure theorem for finitely generated modules over a principal ideal domain (also called Fundamental theorem of finitely
generated modules over a PID in ...
- 77
3
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0
answers
30
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Linearly Compact Module in $R-Mod$
Definition: A module $M$ is called linearly compact if for a family of cosets $\{x_{i}+M_{i}\}_{\triangle}$, $x_{i}\in M$, $\triangle$ is a directed set, and submodules $M_{i}\subset M$ (with $M/M_{i}$...
- 113
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1
answer
43
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Can I use Auslander-Buchsbaum formula for polynomial rings? [closed]
Let $R=k[x,y]$ be a polynomial ring over a field $k$ and set $T=R/\langle xy,y^2\rangle $. Let $f:R\to T$ be the natural ring epimorphism. Is $\operatorname{pd}_RT=2$?
we know that $\...
- 1,244
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When is $\mathbb Z/n\mathbb Z$ semisimple as $\mathbb Z$-module? [duplicate]
I'm looking for some characterization of $\mathbb Z/n\mathbb Z$ being semisimple as $\mathbb Z$-module. Obviously for $n$ prime this is the case, as $\mathbb Z/n\mathbb Z$ is simple under that ...
- 107
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answers
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Classification of 4 dimensional real associative unital algebra
I think I have a complete list for all the commutative ones, maybe with possible repeats (I did try my best to make sure none are same up to isomorphism):
$\mathbb{R}^4 \simeq \begin{bmatrix}a&0&...
- 193
1
vote
1
answer
48
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submodule of completely reducible module is completely reducible
I attempted to solve problems from the textbook 'Basic Abstract Algebra' by P.B. Bhattacharya, S.K. Jain, and S.R. Nagpaul, specifically Chapter 14, Section 4, but encountered difficulties in ...
- 97
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0
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29
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The name for a type of map between vector spaces
Is there a name for a map $f:V \to W$ between two $\mathbb{K}$-vector spaces that is not linear map but which still staisfies
$$
f(\lambda v) = \lambda f(v), ~~~~~ \textrm{ for all } \lambda \in \...
1
vote
1
answer
33
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For $\varphi : \bigoplus_I R \rightarrow M$ the canonical projection, when is $\varphi^{-1}(m)$ finite for each $m \in M$?
For any $R$-module $M$, we can find some set of generators $S$ and construct a surjective map $\varphi : \bigoplus_S R \rightarrow M$ by sending the basis elements of $\bigoplus_S R$ to the generators ...
- 669
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Construct a short exact sequence that is not split
I am currently studying about exact sequence. To illustrate that not all short exact sequences are split, my teacher provided an example in the homework.
For every $m \geqslant2$ construct a short ...
- 49
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42
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Question on Proof of Splitting Lemma for Modules
The setup of the question is as follows. Let the following be a short exact sequence of modules:
$$
0 \rightarrow A \xrightarrow{i} B \xrightarrow{p} C \rightarrow 0,
$$
and let $r: B\to A$ be a map ...
- 2,102
3
votes
1
answer
38
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Noetherian, self-injective ring $R$ with non-torsionless $R$-module
Let $R$ be an (associative, unital) ring. By an $R$-module we mean a left $R$-module. We call an $R$-module torsionless if it can be embedded into a direct product of the regular $R$-module $R$.
I am ...
- 1,055
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votes
1
answer
67
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Overview for properties of modules
I was wondering if there exists a nice overview for properties of modules like being cyclic, simple, semisimple, indecomposable, free, noetherian, artinian and having finite length. I'm thinking about ...
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4
answers
217
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Every $K[G]$-module is torsionless?
Let $G$ be a finite group and $K$ a field. Consider the group ring $R:=K[G]$. Let $M$ be a (left) $R$-module. Is it true that then there exists a set $S$ and an injective $R$-module homomorphism
$M\...
- 1,055
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votes
1
answer
45
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How to understand Free Module $K[x]^r=\bigoplus_{i=r}^rK[x]e_i$
How can i understand of the free module $K[x]^r=\bigoplus_{i=r}^rK[x]e_i$ where $e_i=(0,\ldots ,1, \ldots 0) \in K[x]^r $ denotes the i–th canonical basis vector of $K[x]^r$.
We call $x^\alpha e_i=(0,...
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Every finitely generated module is sum of local modules
Note: All modules are over a $K$-algebra $A$ with $K$ a field and the underlying ring of $A$ is unital (but not necessarily commutative).
Definition: A module $V$ is local if there is a maximal ...
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1
answer
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Free module over a countable set [closed]
Suppose $R$ is a commutative ring. Is $R^{\oplus \mathbb{N}}$ isomorphic to $R^{\oplus \mathbb{N} }\oplus R^{\oplus \mathbb{N}}$ as $R$-modules? If so, how do I find an explicit isomorphism?
Edit: I ...
- 25
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1
answer
47
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Why direct sum of modules admits canonical projections?
This is more of a moral (i.e. category theoretical) question.
In the category of $R$-modules for a ring $R$, the product is the direct product $M=\prod_{i\in I}M_i$ with canonical projections $\pi_i\...
- 438
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0
answers
26
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The Grassmann connection is a connection
Let $\mathcal{A}$ be a *-algebra and $p\in M_N(\mathcal{A})$ an orthogonal projection.
I need to show that $\nabla=p\circ d$ defines a connection on $\mathcal{E}=p\mathcal{A}^N$, where $d$ is acting ...
2
votes
1
answer
55
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$\mathbb{Q},\mathbb{R}$ and $\mathbb{C}$-vector space isomorphism from $\mathbb{Z}$-module isomorphism.
Let $A,B$ be $\mathbb{C}$-vector space. We can view them as a $\mathbb{Z}$-module. Suppose that there is a $\mathbb{Z}$-module isomorphism $\phi$ between $A$ and $B$. Then can we have a natural $\...
- 21
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1
answer
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rank of a free submodule of a free module of infinite rank
I am currently studying the free modules and I am stuck in the following question. Please help me.
I know that if $M$ is a free module on an infinite subset $A$ over a ring $R$ (not necessarily ...
- 368
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39
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Understanding the free $R$-algebras.
Let $R$ be a commutative ring, $R\left< x,y \right>$ the free $R$-algebra on indeterminates $x$ and $y$. If $z_i=xy^i$, I want to show that the $R$-subalgebra generated by the elements $z_i$ is ...
- 401
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28
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Doubt concerning the definition of the tensor product of modules
Let $R$ be a ring with unit and consider a right $R$-module $M$ and a left $R$-module $N$. The tensor product $(M \otimes_R N, \otimes_R )$ of $M$ and $N$, is usually defined as the quotient of the ...
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Reference Request: Extension of Scalars, Free and Projective Modules
In the comments of this question:
What does it mean when the extension of scalars is free?,
it is mentioned that if $ \psi: R \rightarrow S $ is a faithfully flat ring homomorphism between commutative ...
- 11
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0
answers
31
views
Automorphism group of an R-module
Let R be a commutative ring.
It is well known that the automorphism group of the module $R^n$ is isomorphic to $GL_n(R)$. Is their a way to measure how much this fails for an arbitrary module M? ...
- 733
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votes
1
answer
46
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Indecomposable $K[T]$-modules with $\mathrm{char}(K)=2$
Let $K$ be a field of characteristic $2$. Let $G = \mathbb Z / 2\mathbb Z$. The goal is to determine all indecomposable $KG$-modules up to isomorphism.
My main problem is to show that such a module is ...
- 438
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votes
0
answers
21
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Submodule of $\mathbb{Z}$-module with the same rank
For a $\mathbb{Z}$-module $M$ in the field of algebraic numbers, suppose that it has a $\mathbb{Z}$-submodule $N$ which has the same rank with $M$. Shall one deduce that $M=N$? Or can anyone please ...
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0
answers
26
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If two integer matrices $A$ and $B$ have the same row space, $A=UB$ for some unimodular matrix $U$?
I am trying to prove that if $A$ and $B$ are matrices over the integers of the same dimension $m\times n$ and they have the same row space (that is, the set of all linear combinations with integer ...
- 874
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1
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53
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Functor from a filtered category to the category of R-Mod
The following result appears in Charles Weibel's book An introduction to homological algebra as lemma 2.6.14 in page 56.
Lemma Let $I$ be a filtered category and let $A:I\rightarrow R\text{-mod}$ be a ...
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1
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Example request of a finite dimensional algebra of infinite global dimension with a module with no self-extensions
I'm looking for an example of a finite dimensional algebra of infinite global dimension with a non projective module $X$ with no self-extensions, that is $\text{Ext}_A^i(X,X) = 0$ for $i>0$. The ...
- 65
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votes
1
answer
29
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relation between support and submodules
Support of an $R$-module $M$ is defined as $\mathrm{Supp}_R(M)=\{\mathfrak{p}\in Spec(R); M_{\mathfrak{p}}\neq 0\}$. Let $M$ and $N$ two finitely generated modules over a Noetherian ring $R$ such that ...
- 381
3
votes
1
answer
71
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Infinite Product of Module categories over rings is not equivalent to Module category over infinite product of rings
I am looking for a counterexample to the following:
Let $(A_i)_{i = 1}^{\infty}$ be an infinite family of rings with unity. Let $\text{Mod}A$ denote the category of right $A$ modules for a ring $A$.
...
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