Questions tagged [modules]
For questions on modules over rings.
704 questions
5
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The centralizer $Z_{(n-1,1)} = Z[\mathbb C[S_n],\mathbb C[S_{n-1}]]$ as a torsion module over $Z\mathbb{C}[S_{n-1}]$
Let $ S_n $ denote the symmetric group on $n$ letters, and $ \mathbb{C}[S_n] $ its group algebra.
Let $X_n$ be the $n$-th Jucys–Murphy element $X_n = \sum_{k=1}^{n-1} (k\ n)$.
Denote by $Z_n = Z(\...
- 453
7
votes
1
answer
372
views
How to do this computation inside a free module over a connected ring constructively?
Let $k$ be a commutative ring. For a set $M$, denote by $k[M] = \bigoplus_{m \in M} k \cdot m$ the free $k$-module generated by the elements of $M$.
In the end, my question will be about constructive ...
- 1,648
2
votes
2
answers
215
views
Examples of infinite length rings that admit a faithful module of finite length
What are examples of rings $R$ (associative with unit) that have the following property? There is a faithful left $R$-module $X$ of finite length and $R$, as a left $R$-module, has infinite length. ...
- 844
3
votes
1
answer
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Rings for which every module of projective dimension at most 1 is projective
Let $R$ be an associative ring with identity.
Recall that a pair of classes of modules $(\mathcal{A}, \mathcal{B})$ in $R\text{-Mod}$ is called a cotorsion pair if
$$
\mathcal{A} = {}^{\perp}\mathcal{...
- 257
2
votes
0
answers
97
views
When does a module homomorphism on a product module into the ground ring depend only on finitely many coordinates?
Obviously we assume the axiom of choice as it is obviously true.
Consider a commutative unital ring $R$ with $0_R\neq 1_R$.
The set $R^\mathbb{N}$ of all functions from $\mathbb{N}$ to $R$ is an $R$-...
- 1,138
1
vote
0
answers
53
views
Indecomposable modules over a monomial algebra
I has a question about indecomposable modules over monomial algebras.
An admissible ideal $I$ of a path algebra $kQ$ is called monomial if it is generated by some paths of length at least two. The ...
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votes
3
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605
views
Motivations and applications of torsion theories
I have recently observed a significant number of published papers that explore torsion theories or torsion pairs on various categories.
Upon researching the subject, I found that torsion theories have ...
- 77
3
votes
1
answer
213
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How can one identify certain modules under an isomorphism of algebras?
Let $g:=(1,2,3)$, let $h:=(4,5,6)$, and let $G:=\langle g\rangle \times \langle h\rangle \cong C_3\times C_3$.
Let $p:=3$ and let $k:=\overline{\mathbb{F}_p}$.
Let $Q:=$
$\ \ \ \ \ \ {}^\bullet$
$a \...
- 317
3
votes
2
answers
292
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Examples of representations of finite groups with conditions on the tensor square
I would like to have examples of a finite group, $G$, with a finite dimensional representation, $V$, (over the complex numbers, say) with four conditions:
$V$ has a $G$-invariant inner product
The ...
- 570
2
votes
0
answers
147
views
Generic modules over finitely presented algebras of infinite representation type
Let $k$ be an infinite field and $A$ a $k$-algebra (associative with unit) such that there are infinitely many non-isomorphic finite dimensional indecomposable (left) $A$-modules. Consider the ...
- 844
5
votes
0
answers
497
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Structure of induced module of $\mathrm{SL}_2(\mathbb{C})$
$\DeclareMathOperator\SL{SL}$Let $G=\SL_2(\mathbb{C})$, $T$ be the diagonal matrices, $B$ be the upper triangular matrices, and $U$ be the strictly upper triangular matrices.
Let $\theta$ be a ...
- 1
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0
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113
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Nilpotent group acting on abelian $p$ groups / modules over the Heisenberg group algebra
This is a follow-up to a post on Math Stack Exchange. Thanks to Corentin for the ideas and reference in the previous question.
Let
$$
H = \langle x, y \mid [[x,y],x] = [[x,y],y] = 1 \rangle
$$
be the ...
- 1,063
4
votes
1
answer
353
views
How can one describe subcategories of R-modules that are both reflective and coreflective?
Do there exist any general "descriptions" of exact abelian (aka weak Serre) subcategories $A$ of $R$-Mod such that the (full) embedding $A\to \text{$R$-Mod}$ possesses both a left and a ...
- 17.5k
1
vote
1
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174
views
Iwasawa module and inverse limit
Let $\Lambda=\mathbb{Z}_p[[\Gamma]], \Gamma \cong \mathbb{Z}_p$ and $M$ a compact $\Lambda$-module. Do we always have the equation?
$$
M \cong \varprojlim_n\left(\mathbb{Z}_p\left[\Gamma / \Gamma^{p^...
- 463
2
votes
0
answers
105
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How to compute $\operatorname{Ext}^1$ over matrix rings and modules of the form $R/A^nR$?
Over $\mathbb{Z}$, it is classical that
$\operatorname{Ext}^1_{\mathbb{Z}}(\mathbb{Z}/n\mathbb{Z},\, \mathbb{Z}/m\mathbb{Z})
\;\cong\; \mathbb{Z}/\gcd(n,m).$
I would like to understand how this ...
- 257
3
votes
0
answers
186
views
Modules over two-variable polynomial rings
Let $k$ be a field, and $k[x,y]$ be the polynomial ring in two variables. Feel free to assume $k$ is algebraically closed.
Are there any general structural results about finitely generated infinite-...
- 41
4
votes
0
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264
views
Heisenberg group acting on infinitely generated torsion abelian groups
I copied this question from Math Stack Exchange in the hope that some experts on finitely generated soluble groups or modules over non-commutative rings can offer hints for constructing an example of ...
- 1,063
3
votes
1
answer
320
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Generalization of a well-known linear algebraic result to modules
There is a well-known result in linear algebra that is stated below:
Suppose that $g, f_{1}, f_{2}, \ldots, f_{r}$ are linear functionals on a vector space $V$ and let $N, N_{1}, N_{2}, \ldots, N_{r}$ ...
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1
vote
0
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134
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Module Algebra and Smash Product
$\newcommand\hash{\mathbin\#}$I am reading the paper "E. Kirkman, J. Kuzmanovich, and J. J. Zhang. “Gorenstein Subrings of Invariants under Hopf Algebra
Actions”. In: Journal of Algebra 322 (2009)...
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votes
3
answers
952
views
Does group action on a ring induce an action on a Morita equivalent ring
Recall that we say two unital rings $A$ are $B$ are Morita equivalent if the category of left $A$-modules $A$-Mod is equivalent to the category of left $B$-modules $B$-Mod.
Now if a group $G$ acts on $...
- 9,297
0
votes
0
answers
114
views
Finite length module over a simple noetherian ring
Let M be a finite length module over a simple noetherian domain. If M/rad(M) is cyclic does it follow that M is cyclic.
- 376
3
votes
1
answer
135
views
If every direct summand of a module $M_R$ has a unique complement, then every summand is fully invariant
Let $R$ be a ring with unity and $M$ a right $R$-module. Assume that every summand of $M$ has a unique complement, i.e. $M=A\oplus B =A \oplus C \Longrightarrow B=C$. Does this imply that every ...
- 406
0
votes
1
answer
111
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A reverse choice of a norm on a finite-dimensional Banach module
$\newcommand{\norm}[1]{\left\lVert#1\right\rVert}$In the following, everything is finite-dimensional over a base field $\mathbb{K} \in \{\mathbb{R},\mathbb{C}\}$.
Let $(A,\norm{\cdot}_A)$ be an ...
- 7,933
0
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0
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89
views
Submodules of a free module over a finite local PIR
A duplicate of this:
Let $R$ be a finite local principal ideal ring that is not a field (I'm going to use $\mathbb{Z}/p^n\mathbb{Z}$ as an example, in the general case the only proper ideals are the ...
- 327
2
votes
0
answers
169
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When is module category determined by its subcategory of finitely generated modules?
I came across an exercise in Anderson, Fuller [Exercise 22.4] that claims
For two rings $R$ and $S$, their categories of (all) left modules are equivalent if and only if their left categories of ...
- 121
3
votes
1
answer
252
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Proving the intersection of lattices is finitely generated over non-discrete valuation ring
I am trying to loosely follow Casselman's "The Bruhat-Tits Trees of SL(2)" instead using the field $F=\mathbb R_\rho$, a quotient of a subring of the hyperreals. It has a non-archimedean ...
- 131
4
votes
0
answers
123
views
When is every $s$-pure exact sequence pure, and every $s$-flat module flat?
Let $R$ be a ring. Consider a short exact sequence of left $R$-modules:
$$
\eta: \quad 0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0 .
$$
We say that $\eta$ is $s$-pure if it remains exact ...
- 257
7
votes
1
answer
375
views
The second Brauer-Thrall conjecture for finitely generated algebras
Let $k$ be an infinite field and $A$ a finitely generated $k$-algebra, so $A \cong k \langle x_1, \dots, x_n \rangle / I$. We say that $A$ is of
infinite type if there are infinitely many finite ...
- 844
0
votes
1
answer
119
views
Induced map on Tor via a canonical projection between cyclic groups
In a homological algebra course, I encountered the following claim:
Let $p$ be a prime number. Consider the cyclic groups:
$C_{p^2}=\mathbb{Z} / p^2 \mathbb{Z}$,
$C_p=\mathbb{Z} / p \mathbb{Z}$, and ...
- 257
0
votes
0
answers
120
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Example of a morphism of $\mathbb{Z}$-modules that kills maps from simples but not from some finitely presented module
I am trying to construct an example of a morphism of $\mathbb{Z}$-modules
$$
f: M \longrightarrow N
$$
satisfying the following properties:
$f$ is nonzero,
$M$ and $N$ are not projective (i.e., not ...
- 257
1
vote
1
answer
202
views
When is every two-sided maximal ideal idempotent?
Let $R$ be a (possibly noncommutative) ring.
Recall that an ideal $I \subseteq R$ is called idempotent if $I^2=I$, meaning that every element of $I$ can be written as a finite sum of products of ...
- 257
10
votes
1
answer
337
views
A relaxed definition of "invertible matrix" in $\mathbb{Z}/p^n\mathbb{Z}$
One of the possible definitions of invertible matrix over a field is the following.
Def. 1. Let $\mathbb{F}$ be a field and let $\mathbf{A}$ be a $k \times k$ matrix over $\mathbb{F}$. Then $\mathbf{A}...
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3
votes
1
answer
188
views
How to characterize morphisms $f: M \rightarrow N$ such that $\operatorname{Tor}_1(S, f)=0$ for all simple right $R$-modules $S$?
Let $R$ be a ring (not necessarily commutative), and let $f: M \rightarrow N$ be a morphism of left $R$-modules. We say that $f$ is an S -morphism if for every simple right $R$-module $S$, the induced ...
- 257
3
votes
1
answer
426
views
Strengthening of Proposition 2.8 in Atiyah-Macdonald
In Introduction to Commutative Algebra (Atiyah-Macdonald), proposition 2.8 reads:
"Let $A$ be a local ring, $m$ its maximal ideal, $k = A/m$ its residue field. Let $M$
be a finitely generated $A$-...
- 327
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votes
0
answers
140
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Is the sum of saturated subsheaves also saturated?
Let $\mathcal{E}$ be a torsion free coherent sheaf on $\mathbb{P}_2$. Then sub-sheaves
$\mathcal{E}_1$ and $\mathcal{E}_2$ of $\mathcal{E}$ are saturated if $\mathcal{E}/\mathcal{E}_i$ are torsion ...
- 31
4
votes
1
answer
162
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Proof that any block of $kG$ with defect groups of order two is Morita-equivalent to $kC_2$
I asked the following on math.stackexchange.com (see https://math.stackexchange.com/questions/5043642/proof-that-any-block-of-kg-with-defect-groups-of-order-two-is-morita-equivalen) but did not ...
- 353
0
votes
0
answers
114
views
inverse limit and filtration
Thanks for your reading. Given an inverse system of rings $\left(A_n, \phi_n: A_n \rightarrow A_{n-1}\right)_{n \in \mathbb{N}}$ where every homomorphism $\phi_n$ is surjective, we study the inverse ...
- 463
0
votes
0
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192
views
Jacobson radical, group ring and kernel
$\DeclareMathOperator\GL{GL}$Let $G$ be a finite group and $F$ be a field of prime characteristic $p$. Suppose further that $G$ has a normal $p$-Sylow subgroup $N$. By the question linked here:
...
- 463
-1
votes
1
answer
326
views
Ring structure of tensor product
Let $R$ be a commutative $\mathbb{Z}_p$-algebra integral closed domain, $\mathcal{O}_n=\mathbb{Z}_p[\zeta_n]$, and $\zeta_n$ be n-th roots of unity. For the tensor product $R \otimes_{\mathbb{Z}_p}\...
- 463
0
votes
0
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103
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Existence of maps between two modules over almost algebras living in an entourage. Lemma 2.3.7 in Gabber, Ramero
I would like to ask about the proof of Lemma 2.3.7 in "Almost Ring Theory" by Gabber and Ramero. Their proof proves part (iv) of the claim and I am specifically having trouble with the line:
...
- 9
3
votes
0
answers
120
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primitive idempotents in semisimple group algebras
Let $G$ be a finite group, and $M$ be a minimal left ideal of $\mathbb{R}G$ (or irreducible $\mathbb{R}$-representation of $G$).
There are three possibilities for $M$:
Case 1: $M \otimes \mathbb{C}$ ...
- 203
2
votes
0
answers
116
views
orthogonality relation in irreducible representations
Let $G$ be a finite gorup. I think the orthogonality relation between elements of $\mathbb{C}$-irreducible representation of $G$ is well-known. If $R$ is any orthogonal realization of an irreducible ...
- 203
0
votes
0
answers
50
views
An example of a left uniform ring with one right ideal not an SIP-module
Recall that a module $M_R$ is called uniform if the intersection of any two nonzero submodules is nonzero. A ring (with $1$) $R$ is called left (resp., right) uniform if the module ${}_RR$ (resp., $...
- 406
5
votes
1
answer
349
views
The classification of finitely generated modules over the ring of (Laurent polynomials in multiple variables)
Let $M=\mathbb Z [x_1^{\pm1},\dots,x_n^{\pm1}]$ be the ring of Laurent polynomials in multiple variables. May I ask if there are known results that give the classification/nice description of the ...
- 1,063
1
vote
1
answer
224
views
Simple modules of the Weyl algebra
Let $\mathbb{F}$ be a field and W be the Weyl algebra, as the algebra over $\mathbb{F}$ generated by $a,b$ with relation $ab-ba=1$.
The description of simple modules over the Weyl algebra over ...
- 477
0
votes
0
answers
52
views
Semiperfect modules are $D1$
Let $M_R$ be a module. We say that $M$ is lifting or has $D1$ if for every submodule $N \subset M$ there is a decomposition $M=A \oplus B$ such that $A \subset N$ and $N \cap B \ll M$ (that is, $N \...
- 406
0
votes
0
answers
129
views
Finitely generated module over a skew Laurent polynomial ring $\mathbb{K}[x_1^{\pm 1},x_2^{\pm1}]$
Let $A := \mathbb{K}_{\Lambda}[x_1^{\pm 1}, x_2^{\pm 1}, x_3^{\pm 1}, x_4^{\pm 1}]$, $B := \mathbb{K}_{\Lambda'}[x_1^{\pm 1}, x_2^{\pm 1}, x_3^{\pm 1}]$, and $C := \mathbb{K}_{\Lambda''}[x_1^{\pm 1}, ...
- 923
2
votes
0
answers
124
views
Does a matrix ring over a ring satisfy the Koethe conjecture if the coefficient ring itself satisfies the Koethe conjecture?
I just want to know whether the following statement is true or false.
If $R$ is a ring satisfying the Koethe conjecture, then the matrix ring over $R$ also satisfies the Koethe conjecture.
Or is it ...
2
votes
2
answers
255
views
A semisimple ring is left square-full iff it is right square-full
Let $R$ be a ring with unity and $M$ any right $R$-module. A submodule $X$ of $M$ is called square-root in $M$ if $X \oplus X$ embeds in $M$ (i.e., there exists a monomorphism $X \oplus X \to M$). ...
- 406
2
votes
0
answers
119
views
When is a bimodule that is projective as a right and as a left module also projective as a bimodule
Are there practical criteria for determining when a bimodule that is projective as a right and as a left module is projective as a bimodule? Some illustrative examples of what goes wrong and what goes ...
