All Questions
195 questions
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Isoartinian and isosimple modules
I'm reading this article by A. Facchini and Z. Nazemian, in wich they discuss modules with chain conditions up to isomorphism. A couple of the main concepts are the following:
Definition
We say that ...
CommunityBot
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votes
1
answer
46
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Equivalent condition for distributive property of $\mathcal{S}(M)$ [closed]
$\textbf{Definitions:}$
Let $M$ be an $A$-module. The collection $\mathcal{S}(M)$ of all submodules of $M$ is partially ordered by the inclusion relation. The collection $\mathcal{S}(M)$ is said to ...
- 5,878
5
votes
2
answers
706
views
Zero tensor product over a complex algebra?
Let $A$ be an algebra over $\mathbb{C}$. Let $M$ be a left $A$-module, let $N$ be a right $A$-module and consider the tensor product $N \otimes_A M$, which is a complex vector space.
Q1: Can this ...
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4
votes
1
answer
616
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When does the forgetful functor from modules to vector spaces have a right adjoint?
Given any algebra $R,$ when does the forgetful functor
$R\text{-}Mod \rightarrow Vec$
have a right adjoint?
Does this imply any finiteness conditions on R?
Is there a book/paper discussing this?
I'...
- 91
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answers
91
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What are all pairs $(R,M)$ of a ring $R$ and a two-sided $R$-module $M$ such that all endomorphisms of $M$ are scalar multiples of $\text{id}_M$?
I was playing with some endomorphism rings and got curious whether there is a classification of all two-sided (not necessarily unitary on any side) modules $M$ over a (not necessarily unital) ring $R$ ...
- 4,713
2
votes
1
answer
310
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Flatness of submodules of free modules
Are submodules of free $\mathbb{Z}[G]$-modules flat? if not what conditions on $G$ makes it true? $G$ is an infinite group.
If $\mathbb{Z}[G]$ is a Prüfer domain then this is true. Can a group ring $...
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4
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1
answer
158
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Dimension of a module over a left-Ore domain
If $R$ is a domain, and $M$ a (left) $R$-module, what are the different notions of dimension of $M$ and their respective assets, what do they measure?
I found out that if $\dim_RM$ is the cardinal of ...
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2
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1
answer
143
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A weak Schur's lemma for non-semisimple finite dimensional algebras
Let $B \subseteq C$ be an inclusion of finite dimensional (associative) algebras over a field $k$. Assume that $C$ is a free $B$-module. Let $\bigoplus_i U_i$ be
a decomposition of $B$ into ...
- 35.2k
5
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187
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Any f.p. faithful simple module over a primitive group ring?
Recall that a ring $R$ is primitive if it has a faithful simple left module. Let $G$ be a countable discrete group and $R=\mathbb{k}G$, where $\mathbb{k}$ denotes some field or $\mathbb{Z}$.
There ...
- 63.9k
13
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1
answer
595
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Inverse of the Structure Theorem for Finitely Generated Modules over PID
We know that for a PID $R$, any finitely generated module is of the form $\frac{R}{(a_1)} \oplus \dots \oplus \frac{R}{(a_s)} $.
I was wondering if the converse of this statement is true, that is, is ...
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151
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Lifting idempotents and projective coverings --- reference request
Let $R$ be a (non-commutative, unital) ring with Jacobson radical $J$, write $\overline{R}=R/J$ and denote the quotient map $R\to \overline{R}$ by $a\mapsto \overline{a}$.
It is easy to see that if ...
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7
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1
answer
242
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Injective indecomposable modules over Laurent polynomial rings
What does the injective envelope of $\mathbb C[x,x^{-1}]/(p(x,x^{-1}))$ as a $\mathbb C[x,x^{-1}]$-module look like where $p(x,x^{-1})$ is an irreducible element? I’m sure this is well known, but ...
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answer
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Description of modules over self-injective algebras of finite representation type
Is there any description of indecomposable modules and irreducible morphisms over self-injective algebras of finite representation type? I am interested mainly in such a description for nonstandard ...
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1
answer
780
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Equivalence of idempotents and projective modules over nonunital rings
For a nonunital ring $R$ (or "rng") one has to be a little bit careful when considering the category of (left or right) $R$-module and, furthermore, the standard equivalent definitions of projective ...
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Given a representation-infinite algebra, when is every AR component infinite?
Let $A$ be a finite dimensional algebra over an algebraically closed field $K$. The Auslander-Reiten quiver $\Gamma_A$ of $A$ is a means of presenting the category of finitely generated right $A$-...
- 482
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1
answer
111
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Extensions of modules of type $FP_n$
Let $R$ be a ring (not necessarily commutative, but with a unit). Recall that an $R$-module $M$ is of type $FP_n$ if $M$ has a partial projective resolution of length $n$ whose terms are all finitely ...
- 167
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473
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Why do we want $p$-permutation modules in splendid equivalences?
First Rickard (in Splendid Equivalences: Derived Categories and Permutation Modules ) and then Rouquier (Block theory via stable and Rickard equivalences, Appendix A.1) define splendid equivalences ...
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1
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Adjoints of scalar extension and scalar coextension
Let $h\colon R\rightarrow S$ be a morphism of commutative rings. We consider the following functors (I am aware that the notations may be different in other contexts):
$h^*$: Scalar extension by ...
- 35.2k
14
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1k
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Kaplansky's theorem and Axiom of choice
Kaplansky in his paper titled by Projective Modules gave an important and essential theorem as follow:
Theorem: Let $R$ be a ring, $M$ an $R$-module which is a direct sum of (any number of) countably ...
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3
answers
446
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Is the category of symmetric bimodules over a commutative ring closed under extensions?
Let $A$ be a commutative ring, and consider the category of bimodules over $A$.
An $A$-bimodule $M$ is called symmetric if $a\cdot m = m \cdot a$ for all $a \in A$, $m \in M$.
Is the category of ...
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4
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2
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664
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Modules over infinite rings which can not be a finite union of their proper submodules
It is well known that a vector space over an infinite field cannot be a finite union of its proper subspaces.
Does this fact have an immediate and obvious generalization to modules over infinite ...
- 14.6k
14
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2
answers
618
views
"Inner product" between prime factorizations
Let $x \in \mathbb{Q}^+$. Then $x$ can be expressed uniquely as a product over primes:
$$x = \prod_{p \text{ prime}} p^{\nu_p(x)}$$
where $\nu_p(x)$ is the p-adic valuation of $x$. Let $\nu(x) = \...
- 53.3k
1
vote
1
answer
124
views
A nonreduced quotient ring
I am searching for a commutative ring $R$ and a semisimple $R$-module $M$ such that the quotient ring $\frac {R}{soc(R)\cap ann_R(M)}$ has a nonzero nilpotent element. Here, $soc(R)$ means the socle ...
- 4,713
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answers
110
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Hochster-Roberts theorem
I am trying to understand this theorem, and as an example I try this case: Let $R=\mathbb{C}[z_1^{\pm},\ldots,z_n^{\pm}]^{S_n}$ be the algebra of Laurent polynomials. Now $R$ should be somehow ...
2
votes
1
answer
243
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Isomorphism of irreducible R-modules
Let $R$ be a $k-$algebra and $M,N$ two irreducible $R-$modules, isomorphic as vector spaces. If we know that for every $r\in R$ we have the same eigenvalues on $M$ and $N$ (with multiplicities) is it ...
- 4,713
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indecomposable module over a local ring
I ask this in mathematics for some days.it doesn't have an answer up to now. https://math.stackexchange.com/questions/2565828/indecomposable-module-over-a-local-ring
As we all know, for an arbitrary ...
- 35.2k
13
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1
answer
670
views
When is $A\otimes R$ a free $R$-module?
Let $R$ be a commutative ring. If I am not mistaken, there is the following fact:
For a finitely generated abelian group $A$, the $R$-module $A\otimes R$ is free if and only if we can write the ...
- 7,893
1
vote
1
answer
149
views
Finding modules to check for finite global dimension
Given a finite dimensional algebra $A$ and a generator-cogenerator $M$ and let $B:=End_A(M)$. $B$ has finite global dimension iff every $A$-module has finite $add(M)$-resolution dimension, which is ...
- 356
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294
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Is it true that the functor of completion of a module over a local ring is injective on isomorphism classes?
Let $A$ be a commutative Noetherian local ring and $\hat A$ be its completion. Then we have the functor of completion from the category of finitely generated $A$-modules to the category of finitely ...
- 1,484
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1
answer
302
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Endomorphism ring of bimodules
Given an algebra $A$ with a right $A$-module $M$ with $End_A(M) \cong A$.
Then we can view $M$ as a natural $A$-bimodule. When is $M$ as a bimodule indecomposable and what is its endomorphism ring as ...
- 9,650
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335
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Graded and projective (but not bounded below) module that is not graded-projective?
Let $A = \bigoplus_{n = 0}^\infty A_n$ be a graded algebra over a field $k$ that is locally finite: each $A_n$ is a finite-dimensional $k$-vector space. We say that a graded left $A$-module $P = \...
- 6,700
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556
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In modules over finite rings $Rv = Rw \iff R^\times v = R^\times w$
Let $R$ be a finite ring (with unit, possibly non-commutative), and $M$ a left module over $R$. Let $v,w\in M$. Then
$$Rv = Rw \iff R^\times v = R^\times w.$$
This follows from Lemma 6.4 in Hyman ...
- 11.6k
2
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0
answers
214
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"Strongly finite rings" or similar term for a condition implying IBN
While (once again) perusing T.Y. Lam's excellent GTM 189 "Lectures on Modules and Rings" I compared the various conditions given that typically imply IBN, e.g. the (left) strong rank condition, stably ...
- 4,679
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2
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512
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A question with simple and indecomposable modules
Assume $M$ is both noetherian and artinian and fix $S_0\subseteq M$ a simple submodule. How to prove that $S_0$ is contained in some indecomposable direct summand of $M$?
CommunityBot
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self-cogenerator rings
Let $\mathbb{U}$ be a non-empty set (class) of objects of a
category $C$. An object $B$ in $C$ is said to be cogenerated by
$\mathbb{U}$ or $\mathbb{U}$-cogenerated if, for every pair of
distinct ...
- 9,229
2
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1
answer
172
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Morphisms in K-theory: comparison of two pictures
Algebraic $K_0$ group for an algebra $A$ may be defined in terms of stable isomorphism classes of idempotents in $M_n(A)$ or equivalently in terms of isomorphism clasess of finitely generated ...
- 111
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263
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Nonnegatively graded algebra $A$ finitely generated as $k$-algebra iff $A_0$ finitely generated, $A_{>0}$ finitely generated as $A$-module?
This is related to my question here. My question is as follows. How do I see that a nonnegatively graded algebra $A$ is finitely generated as a $k$-algebra if and only if $A_0$ is finitely generated ...
CommunityBot
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68
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On finite Uniform (Goldie) dimensions
1) Is there any characterization of $\Bbb{Z}$-modules with finite uniform dimensions?
2) Find two $\Bbb{Z}$-modules $N, M$ such that $N\leq M$ and $M\hookrightarrow N$ but $N$ is not isomorphic to $M$...
CommunityBot
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3
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378
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$I=mI$, when I is not finitely generated
Let $(R,m)$ be a commutative local ring with unit. Suppose $I$ is an ideal (not finitely generated). If $I=mI$, what can we say about $I$? Can we conclude something interesting?
If $I$ were finitely ...
CommunityBot
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394
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Classification of indecomposable modules in tame hereditary algebras
An algebra $A$ is said to be tame if the isomorphism classes of indecomposable $A$-modules in each dimension occur in a finite number of 1-parameter families. $A$ is said to be of finite ...
- 3,686
1
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1
answer
357
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the relation between projective and quasi-projective modules
An $R$-module $M$ is called quasi-projective if $\text{Hom}_R(M,M)\to\text{Hom}_R(M,N)$ is surjective for every surjective homomorphism $M\twoheadrightarrow N$.
What are the rings $R$ for which every ...
- 35.2k
6
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1
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302
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Why is $\operatorname{nr}_{F[G]}:K_1(F[G])\to Z(F[G])^\times$ a bijection?
Let $A$ be a finite dimensional semisimple $F$-algebra and $K_1(A)$ the Whitehead group of $A$.
By splitting $A$ into its Wedderburn components, the reduced norm map $\operatorname{nr}_A:K_1(A)\to Z(...
- 5,382
1
vote
1
answer
271
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Rank of a locally free $\mathbb Z[G]$-module
This is a basic question. Let $G$ be a finite group, $M$ a finitely generated $\mathbb Z[G]$-module so that the $\mathbb Z_p[G]$-module $M_p$ is free for all prime numbers $p$, i.e. is locally free.
...
- 5,039
6
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1
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A Hom-Tensor identity - $\text{Hom}_{R}(P,B)\otimes _SC \cong \text{Hom}_{R}(P,B \otimes_S C) $
let $R,S$ be associative algebras over $\mathbb{C}$. Let $\mathcal{C} \subseteq$ $R$-Mod be a full abelain subcategory of $R$-Mod which is the category of $R$-modules. Let $B$ and $C$ be, a $(R,S)$-...
CommunityBot
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0
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85
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An order in $\mathbb Q[G]$ which is a maximal $\mathbb Z_p$-order in $\mathbb Q_p[G]$ for finitely many primes $p$
Let $G$ be a finite group and $S$ a finite set of prime numbers. I know that every separable $\mathbb Q$-algebra $A$ contains a maximal $\mathbb Z$-order but I wonder if the following is true.
Is ...
- 255
3
votes
1
answer
303
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An invariant submodule of a projective module
This is a basic question (not research level) which has already been asked on SE by someone else but doesn't yet have an answer so I'd like to repost it on MO.
Let $R$ be a commutative ring with ...
- 11.6k
7
votes
1
answer
456
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Hopfian modules
My question is slightly motivated by basic results in linear algebra, for example, that if $F$ is a field then a surjective linear map $F^n \rightarrow F^n$ is injective. More generally, any ...
- 35.2k
1
vote
3
answers
365
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On modules with finite uniform dimension
Is it true that if a module $M$ has finite uniform dimension then the same is true for its homomorphic images ?
- 321
3
votes
2
answers
475
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Rings all of whose torsion modules are cyclic
Let us call a (possibly non-commutative) ring $R$ "very good" if every finitely generated torsion left $R$-module is cyclic. Here is an example of such a ring:
Let $k=\mathbb{C}((t))$ and let $R=k[\...
- 3,702
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1
answer
241
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Cyclic faithfully flat modules
I am looking for an example of a cyclic faithfully flat $R$-module that is not projective. Could someone help me?
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