# Questions tagged [modules]

For questions on modules over rings.

561
questions

4
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### What are the indecomposable modules over $\mathbb{F}_2(C_2\times C_2)$?

Let $C_2$ be the cyclic group of order $2$ and $\mathbb{F}_2$ the field with $2$ elements. Consider the group algebra $A:= \mathbb{F}_2 (C_2\times C_2)$. It is well-known that $A$ has infinite ...

3
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0
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56
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### Modules with special properties

Let $A$ be a finite dimensional algebra and $M$ an indecomposable (right) module with the property that every nilpotent element of $End_A(M)$ annihilates the socle $soc_A(M)$ of $M$. Note that $M$ is ...

0
votes

1
answer

112
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### Semi simplicity over commutative algebras over non-algebraically closed fields

I have already posted this on stackexchange
I have a question:
If k is an arbitrary field then is it true that if $M$ a finite dimensional $k[x, y]$ is semisimple as a $k[x]$ module and also as a $...

3
votes

0
answers

97
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### Are there (co-)homological obstructions to the extendability of a homomorphism?

Let $k$ be a commutative ring and $A \subset B$ an extension of $k$-algebras. Can we associate to this extension a (co-)chain complex so that its (co-)homology $H$ allows for statement such as
"...

2
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0
answers

367
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### Minimal number of generators of kernel of a matrix of polynomials over $\mathbb Q$ and $\mathbb Z$

$\DeclareMathOperator\im{im}$I have asked some related questions before on math.SE here and on MathOverflow here (answered here). This post is self-contained.
Let $R' = \mathbb Q[x_1,\dotsc,x_n]$, and ...

0
votes

0
answers

51
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### Freeness of injective hull of finite module over Gorenstein Artinian ring

This is Lemma 4.1 in Brochard, Khare, Iyengar: Wiles defect and criteria for freeness. I have managed to understand the proof, except for the following part: Why is $F$, the injective hull of a finite ...

1
vote

1
answer

233
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### Constructing a free resolution of a $\mathbb Z[x_1,\dotsc,x_n]$-module using a related free resolution of a $\mathbb Q[x_1,\dotsc,x_n]$-module

I have asked a related question on math.SE here, but the notation is a bit different.
As the title says, I am interested in constructing a finite free resolution of a $\mathbb Z[x_1,\dotsc,x_n]$-...

1
vote

1
answer

48
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### Hilbert - Samuel multiplicity of $B$ when there is a surjection $A \rightarrow B$

The setting is: Let $A, B$ be commutative, Noetherian, local rings, $\phi:A \rightarrow B$ a surjective homomorphism. Both rings also come with surjections $\lambda_A, \lambda_B$ to a DVR $\mathcal{O}$...

1
vote

1
answer

98
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### Pontrjagin dual of modules [closed]

I am not sure whether this question is appropriate to appear here. If not, I apologize for that.
Given an $R$-module $M$, we define its Pontrjagin dual as $M^{\ast}=Hom_{\mathbb{Z}}(M, \mathbb{Q/Z})$. ...

2
votes

0
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125
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### $\operatorname{Ext}^1$ isomorphic to quotient ring

I'm reading this paper: Brochard, Iyengar and Khare: Wiles defect for modules and criteria for freeness. In lemma 4.5, there is an isomorphism $\operatorname{Ext}_A^1(k,A) \cong \frac{I_A}{\varpi I_A}
...

2
votes

1
answer

75
views

### Nondegenerate pairings versus perfect pairings for finitely generated projective modules

Let $R$ be a (not necessarily commutative) ring, $M$ a left $R$-module, and $N$ a right $R$-module. We say that a pairing
$$
\langle -,-\rangle:M \otimes_R N \to R
$$
is non-degenerate if, for all $n \...

0
votes

0
answers

73
views

### $\operatorname{Ext}$-group in the category of modules versus in the subcategory of finitely generated ones

I am trying to refine my understanding of derived categories.
Let $\text{Mod}_R$ and $\text{Mod}^f_R$ be respectively the categories of modules and finitely generated modules over a Notherian ring $R$ ...

1
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0
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35
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### A bi-variate polynomial interpolation question

Let $R$ be a commutative unital ring, and $R^{m\times k}$ denote the set of $m\times k$ matrices with entries from $R$. A matrix $U\in R^{m\times m}$ is elementary if $U$ is obtained from the identity ...

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0
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52
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### Prove that $f(M)=f^2(M)$ implies $f(M)$ is a direct summand of $M$ whenever $\text{End}_R(M)$ is a reduced ring

Let $M$ be a right $R$-module with the property that every homomorphism $\gamma:Sf\to M, f\in S=\text{End}_R(M)$, extends to $S\to M$. If $S$ has the property $f^2=0$ implies $f=0$ for every $f\in S$...

4
votes

1
answer

152
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### Absolutely irreducible representation and splitting field

Let $A$ be a finite-dimensional algebra over a field $F$. A representation $M$ of $A$ is called absolutely irreducible if $M\otimes_FE$ is irreducible as a representation of $A\otimes_FE$ for all ...

7
votes

0
answers

232
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### Split epimorphism of modules - does the finite case imply the infinite case?

Let $k$ be a field, $A$ a finite dimensional $k$-algebra, $X$ a finite dimensional indecomposable (left) $A$-module and $M$ an infinite dimensional (left) $A$-module. Suppose further we have an ...

1
vote

1
answer

64
views

### Epimorphism going out of an inverse limit into a finite dimensional module

Let $k$ be a field and $A$ a finite dimensional $k$-algebra. Given a sequence of inclusions $M_1 \subseteq M_2 \subseteq \dots$ of $A$-modules consider the direct limit $M:= \bigcup_{i=1}^\infty M_i$. ...

1
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0
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57
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### Is it possible to describe a $k$-basis for $M\otimes_{kH}N$ when $M$ is a $k[G\times H]$-module and $N$ is a $k[H\times K]$-module?

Suppose $k$ is a field for Let $M$ be a finitely-generated a $k[G\times H]$-module and let $N$ be a finitely-generated $k[H\times K]$-module. Then in particular, $M$ and $N$ are finite-dimensional $k$-...

1
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0
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54
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### When some idempotent ideals belong to the Gabriel filter of ideals for a hereditary torsion theory

Let $\mathscr{I}_\sigma$ be the Gabriel filter of ideals for a hereditary torsion theory $\sigma$ over a commutative ring $R$. I am looking for equivalent conditions on either $\sigma$ or $R$ under ...

2
votes

0
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97
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### modules over principal ideal rings

Let $R$ be a commutative principal ideal ring (not necessarily Artinian) and let $M$ be a finitely generated $R$-module. Is $M$ a direct sum of cyclic $R$-modules? (i.e. a generalization of the theory ...

2
votes

1
answer

101
views

### Structure theorem for finitely generated $\Lambda$-modules - uniqueness part

In Iwasawa theory, one of the fundamental results is the following structure theorem for finitely generated modules over the ring $\Lambda = \mathbf{Z}_p[[T]]$.
If $M$ is a finitely generated torsion ...

3
votes

1
answer

181
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### Split monomorphisms of modules - does the finite case imply the infinite case?

Let $k$ be a field, $A$ a finite dimensional $k$-algebra, $X$ a finite dimensional indecomposable (left) $A$-module and $M$ an infinite dimensional (left) $A$-module. Further $X\subseteq M$ and for ...

1
vote

0
answers

74
views

### Exterior algebra of free modules over Hopf algebras

Let $H$ be a commutative, cocommutative Hopf algebra over a field $\mathbb{K}$, and $M$ a free Hopf module over $H$. Is the exterior algebra $\Lambda^k_\mathbb{K} M$ with the diagonal $H$-action
$$h \...

2
votes

1
answer

145
views

### On some claims on cyclic modules over Hecke algebra used in Serre's "Quelques applications du théorème de densité de Chebotarev"

I have been reading section 7 of Serre's "Quelques applications du théorème de densité de Chebotarev" (http://www.numdam.org/item/PMIHES_1981__54__123_0/), and in particular have been trying ...

8
votes

1
answer

399
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### For every ring R, is there a block-diagonal canonical form for a square matrix over R?

This question asks whether there exists an analogue of the Jordan decomposition for an arbitrary ring $R$. This analogue is not necessarily the Jordan-Chevalley decomposition, which is unnecessarily ...

7
votes

2
answers

107
views

### Rings of finite uniserial type

If $R$ is a ring and $M$ an $R$-module, $M$ is uniserial if its lattice of submodules is a chain. Over an Artinian $R$, the chain will be finite. From what I understand, deciding when two uniserial ...

5
votes

0
answers

78
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### Structure of finitely generated $\mathbb{Z}/p^n\mathbb{Z}[[S,T]]$-modules

Let $\Omega=\mathbb{Z}/p\mathbb{Z}[[S,T]]$. $\Omega$ is a commutative, Noetherian and integrally closed domain of Krull dimension 2. According to Bourbaki's commutative algebra VII $\S 4$, if $M$ is a ...

5
votes

0
answers

204
views

### Group cohomology of $\mathbb{Z}$ vs $\mathbb{Z}_p$

Let $M$ be a continuous representation of $\mathbb{Z}_p$ over $\mathbb{F}_p$, likely infinite-dimensional.
There is the inflation map of group cohomology $H^*_{\text{cts}}(\mathbb{Z}_p, M) \rightarrow ...

8
votes

0
answers

245
views

### Matrix decompositions as monoid isomorphisms. Ever considered before?

I've noticed some correspondences between some matrix decompositions and monoid isomorphisms (always to some free commutative monoid), in addition to the one I asked about in a previous question:
...

2
votes

0
answers

93
views

### The "matrix direct sum" monoid modulo unitary equivalence

Given a commutative $*$-ring $(R,*)$, let $M(R,*)$ be the monoid whose elements are matrices over $R$ of all possible shapes and entries, including those that have $0$ columns or $0$ rows. Let the ...

2
votes

0
answers

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### Is this concept of a left-abelian category studied?

A category is abelian if it is preadditive and
it has a zero object,
it has all binary biproducts,
it has all kernels and cokernels, and
all monomorphisms and epimorphisms are normal.
Now we ...

1
vote

0
answers

66
views

### When is a bounded complex of $RG$-modules contractible?

If we have a $p$-modular system $(K, \mathcal{O},k)$, let $R = k$ or $\mathcal{O}$ and $G$ a finite group. When is a bounded complex of $RG-RG$-bimodules $\Gamma$ contractible?
I've seen this response ...

2
votes

1
answer

128
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### When the annihilator of each nonzero submodule is prime

Let $M$ be a fixed faithful $R$-module over integral domain $R$. Is there any equivalent condition (on $R$ or on $M $) under which the annihilator of any nonzero submodule of $M$ to be a prime ideal ...

1
vote

0
answers

37
views

### A "spectral theorem" to SVD reduction for every commutative *-ring

Given any commutative $*$-ring $R$ of uneven characteristic, is it true that for every square matrix $M$ and unitary matrix $W$, if $W^* \begin{bmatrix} 0 & M \\ M^* & 0 \end{bmatrix} W$ is ...

5
votes

1
answer

208
views

### Concept of an exact ideal of a module category

Let $R$ be a ring and $\text{Mod}\,R$ the category of (left) $R$-modules. Consider an ideal $\mathcal{I}$ of $\text{Mod}\,R$. For $R$-modules $X$ and $Y$ let $\mathcal{I}(X,Y)$ be the collection of ...

21
votes

1
answer

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### Reference request: a tale of two mathematicians

I've heard tell the following anecdote involving Pierre Gabriel and Jacques Tit at least twice in a lapse of four years or so:
When P. Gabriel presented the theorem in a conference [sometime around ...

3
votes

1
answer

220
views

### RIng that is flat over a subring as a right module but not as a left module

What is an example of a ring $R$ and a subring $S \subseteq R$ such that $R$ is flat as a right module but not flat as a left module.
The following question is my motivation:
Faithful flatness for ...

4
votes

0
answers

287
views

### A projective module over a domain that is not faithfully flat?

Let $R$ be a (noncommutative) unital ring which is a domain and let $\mathcal{N}$ be a non-zero projective (right) module. Projectivity of course implies that $\mathcal{N}$ is flat, but does the fact ...

2
votes

2
answers

153
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### Module complements to rings embedded in a projective module

Let $R$ be noncommutative unital ring and $M$ a projective (right) $M$-module. Assume that $R$ embedds into $M$ as a right -module.
A) If $R$ is a semisimple ring, then of course $R$ admits an $R$-...

1
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0
answers

77
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### Composition of faithfully flat ring extensions

Let $R$ be a not necessarily commutative, unital, ring, and for simplicity let module always mean right module. We say that a unital ring extension $R \hookrightarrow S$ is flat, or faithfully flat, ...

1
vote

0
answers

92
views

### Finding an injective envelope containing another injective envelope

Let $R$ be a local principal ideal domain (PID) with only two prime ideals $0$ and $P$, and let $M$ be an $R$-module. Let for $r\in R$ and $m\in M$, $rm\not=0$. Now if $E(rm)$ is a fixed injective ...

4
votes

1
answer

287
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### $\|t\| = \sup_{\|z\| \le 1} \|\langle tz,z\rangle\|$ when $t=t^*$

Let $A$ be a $C^*$-algebra, $E$ be a (right) Hilbert $A$-module and $t \in \mathcal{L}_A(E)$ be an adjointable operator satisfying $t=t^*$. Is it true that
$$\|t\| = \sup_{z \in E, \|z\| = 1} \|\...

3
votes

1
answer

176
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### Faithful flatness for rings

Let $R$ be a ring and let $M$ be a right module over $R$. We say that $M$ is faithfully flat as a right module if the functor $M \otimes_R -$ from left $R$-modules to abelian groups that preserves ...

3
votes

1
answer

101
views

### On the definition and an example of silting/tilting subcategories in a triangulated categories according to a paper by Aihara and Iyama

In the paper "Silting mutation in triangulated categories" by Aihara and Iyama, I stumbled upon this nice definition( Definition 2.1) of a tilting/silting subcategory of a triangulated ...

4
votes

1
answer

187
views

### Infinite linearly independent set in finitely generated module

Let $R$ be a (commutative, otherwise the answer is easy, see the comment below) ring and let $M$ be a finitely generated $R$-module. Is it possible that $M$ admits an infinite linearly independent set?...

0
votes

0
answers

49
views

### Tensor products and intersections of modules

Is it true that $A\otimes_{\Bbbk} B \cap B \otimes_{\Bbbk} A = B\otimes_{\Bbbk} B$ if $B \subset A$ are $\Bbbk$-modules over a ring $\Bbbk$?.
It works for $\Bbbk$ a field.
Does it work in any other ...

2
votes

0
answers

91
views

### Is there a category of "chains of modules" that behaves well with taking direct limits?

I came up with the following definition of a category of certain "chains of modules" and want to know if this concept is already known and studied.
Let $R$ be ring. An object in our category ...

2
votes

1
answer

83
views

### Every module of finite uniform dimension is a direct sum of (finitely many) indecomposable

Crossposted on StackExchange on July 28 (no answer so far).
Let $R$ be a (commutative or non-commutative, associative, unital) ring. It is well known that any artinian or noetherian $R$-module $M$ can ...

4
votes

1
answer

160
views

### Is there essentially unique notion of module over monoidal stable $\infty$-categories?

There is this (folklore?) fact: for a commutative ring $R$, the category of $R$-modules is equivalent to the category of internal abelian groups in the slice category $\operatorname{Commutative rings}/...

4
votes

0
answers

122
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### Generalization of the second Brauer-Thrall conjecture for arbitrary Artin algebras

Let $k$ be a field with infinite cardinality and $A$ a finite dimensional $k$-Algebra. The second Brauer-Thrall conjecture states the following: There are infinitely many natural numbers $n_1<n_2&...