Questions tagged [ra.rings-and-algebras]
Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups. For questions specific to commutative algebra (that is, rings that are assumed both associative and commutative), rather use the tag ac.commutative-algebra.
3,365
questions
4
votes
1
answer
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Some folklore about crystaline rings of differential operators
This question is a follow up to my previous question on rings of crystaline differential operators, to which I refer for the adequate definitions.
First, let's consider the case of an algebraically ...
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2
votes
0
answers
89
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An open problem about simple Noetherian rings
The following is a well-known open problem in ring theory (see, for instance, Goodearl, Warfield, 'An introduction to noncommutative Noetherian rings, Appendix, Problem 19)
Question: Let $R$ be a left ...
- 2,733
2
votes
2
answers
401
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Expositions of the classical approach to local class field theory (Brauer group and Hasse invariant)
I've posted this question already on MSE and didn't get much out of it, so I hope it's OK to repost here.
I'm an undergraduate trying to learn local class field theory from the corresponding chapter ...
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5
votes
0
answers
278
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Connections in non-commutative geometry
Let $K$ be a field, $A$ a unital associative $K$-algebra and $M$ a left $A$-module. A connection on $M$ is a $K$-linear map $\nabla:M\to \Omega^1A\otimes_AM$ which satisfies the Leibniz rule. ...
- 401
5
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0
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190
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Rings where all indecomposable modules are projective or injective
Let $A$ be a semi-perfect noetherian ring.
Is there a nice classification of such $A$ such that every indecomposable finitely generated $A$-module is projective or injective?
Im also interested in ...
- 26.1k
3
votes
1
answer
193
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Relation between enveloping algebras and algebras of differential operators
I asked this question on math stack exchange about 3 years ago, but received no answer.
Our base field $\mathsf{k}$ will be algebraically closed of zero characteristic. Let $X$ be an smooth affine ...
- 2,733
2
votes
1
answer
198
views
Wedderburn–Artin like theorem for infinite dimensional Lie algebras?
The Wedderburn–Artin Theorem is one of the cornerstones of the structure theory of (associative) rings.
Wedderburn–Artin Theorem : Let $R$ be a left Artinian ring with zero Jacobson radical. Then $R$ ...
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0
votes
1
answer
90
views
Subalgebras of finite extensions
Suppose that $A\subset B$ is a finite extension of rings. Is it true that every intermediate extension $A\subset C\subset B$ finite over $A$?
- 147
1
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0
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56
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Eigenvalues of a subset of matrix semigroup
My apologies for slightly longer post but I wanted to explain lower dimensional cases and their proofs before asking the actual question, which starts after the phrase The general case below.
A two-...
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6
votes
2
answers
1k
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Prove that $\Bbb C[x,y]/(x^3+y^3-1)$ is not a UFD
I am posting this question on MO since I haven't received any answers on MSE.
Below is my (very elementary) attempt. Feel free to post a solution using facts in algebraic geometry and facts about ...
- 171
4
votes
1
answer
183
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Nonempty intersection of cosets of finite-index subgroups
$\DeclareMathOperator\lcm{lcm}$This question is crossposted from MSE.
Let $H_1,\dots,H_{n+2}$ be cosets of finite-index subgroups of $\mathbb{Z}^n$ and suppose for all $i=1,\dots,n+2$, $\bigcap_{j\neq ...
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3
votes
0
answers
78
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Vershik's conjecture about generic quadratic algebras
Is it still unknown whether very general (lying in a countable intersection of some Zariski opens in corresponding Grassmannian) quadratic algebras $R$ with $\operatorname{dim} R_2 < \frac{3}{4}(\...
- 4,436
1
vote
2
answers
224
views
Link invariants from Hecke relations of higher order
Alexander theorem says oriented links in $\mathbb{R}^3$ can be
represented by closures of braids. Markov theorem says that
braids related by Markov moves produce isotopic braid closures,
and vice ...
- 5,038
0
votes
0
answers
94
views
Is there a "cohomology theory" for involutive algebras?
I'm aware that there are cohomology theories for algebraic structures related to involutive algebras (or "involution algebras" or "$*$-algebras" if you prefer those terms) like Lie ...
- 4,843
3
votes
1
answer
293
views
The associated graded algebra of a finite dimensional algebra
$\DeclareMathOperator\rad{rad}$Let $A$ be a finite dimensional algebra (we can assume that $A \cong KQ/I$, for a quiver $Q$ and an admissible ideal $I$ if that helps).
Denote by $A_G$ the associated ...
- 26.1k
1
vote
1
answer
212
views
Two exact sequences for $R$-modules: does one imply the other?
Consider the following two properties for an $R$-module $M$:
For every endomorphism $f:M\rightarrow M$, there exist central endomorphisms $g, h\in \operatorname{End}_R(M)$ such that the sequence $M_{...
- 321
4
votes
1
answer
411
views
WZW primary correlations in terms of current algebra?
Given the
$\mathfrak{u}(N)$ algebra
with generators $L^a$ and commutation relations
$ [L^a,L^b] = \sum_c f^{a,b}_{c} L^c $ ,
the WZW currents of $U(N)_k$
$$ J(z) = \sum_{n \in \mathbb{Z}} J^a_n z^{-n-...
- 515
1
vote
0
answers
109
views
Degrees of trigonometric numbers
For a rational number $r\in(0,1)$ the number $z=\sin(r\pi)$ is an algebraic number — such numbers appear to be called trigonometric numbers.
What is its degree?
That is, what is the minimal degree of ...
- 8,046
2
votes
0
answers
133
views
$p$-adic Banach group algebra
Let $G$ be a discrete group. Consider the Banach $\mathbb{Z}_p$-algebra: $$c_0(G, \mathbb{Z}_p) = \{ F : G \to \mathbb{Z}_p \mid \lim_{g \to \infty} |F(g)|_p = 0 \}$$ with the product given by the ...
- 1,325
1
vote
0
answers
57
views
A certain subalgebra of $\mathfrak{e}_8$ over a p-adic field
Can $\mathfrak{e}_8(\mathrm{k})$ have a maximal subalgebra isomorphic to $\mathfrak{sl}_1(\mathrm{D})\oplus\mathfrak{g}_2(\mathrm{K})$, where $\mathrm{k}$ is a finite extension of some $\mathbb{Q}_p$, ...
- 2,689
0
votes
0
answers
88
views
Definition of term functions, in universal algebra
According to the definitions in Sankappanavar's universal algebra :
Assume $p$ is a term, then $p(x_1,x_2,...,x_n)$ indicates that the variables occurring in $p$ are among $x_1,...,x_n$. But there is ...
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3
votes
2
answers
305
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Chirality of octonion algebras
Octonion multiplication can be defined with respect to a set of triads. A set of such triads can be represented by a directed Fano plane diagram such as the following two diagrams.
This depicts two ...
-1
votes
1
answer
139
views
Classification of real Clifford algebras
$\DeclareMathOperator\Cl{Cl}$Let $V$ be a real vector space of dimension $p+q$. Let $Q$ be a non-degenerate quadratic form on $V$ of signature $(p,q)$ where $p$ is the number of positive eigenvalues, ...
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2
votes
0
answers
57
views
upper bound for the exponential conjugacy growth rate for non-virtually nilpotent polycyclic groups
Given $n ≥ 0$, the conjugacy growth function $c(n)$ of a finitely generated group $G$, with respect to some finite generating set $S$, counts the number of conjugacy classes intersecting the ball of ...
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7
votes
0
answers
191
views
On the structure of an algebra as a bimodule
$\DeclareMathOperator\End{End}\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\Ker{Ker}\newcommand{\bi}{\mathrm{bi}}\newcommand{\op}{\mathrm{op}}$Let $K$ be a field (say of characteristic zero), and $...
2
votes
0
answers
148
views
The growth rate of the group $\mathbb{Z}[1/2] \rtimes _\phi \mathbb{Z}$, where $\phi (1)$ corresponds to multiplying every number by $2$
Consider the group $G = \mathbb{Z}[1/2] \rtimes _\phi \mathbb{Z}$, where $\mathbb{Z}[1/2] = \{j/2^m \mid j \in \mathbb{Z}, m\in\mathbb{N} \}$, the dyadic rationals, and for every $n\in \mathbb{Z}$, $...
- 763
5
votes
0
answers
275
views
Serre subcategories of the category of chain complexes of modules
Let $k$ be an algebraically closed field of characteristic $0$.
Let $R$ be a commutative $k$-algebra.
We denote by $\operatorname{Mod}(R), C(R), $ and $ D(R)$ the category of $R$-modules, the category ...
- 639
2
votes
1
answer
249
views
Gluing data for modules over a ring with idempotents
Let $A$ be a ring. If $e$ is an idempotent, then there is an abelian recollement involving the categories $A\text{-}\mathrm{Mod}$ and $eAe\text{-}\mathrm{Mod}$. This is Example 2.7 in Homological ...
- 1,021
5
votes
2
answers
388
views
Algebra with three anti-commutator relations
Let $u,v,w \in \mathbb{F}_p^{\times}$. Consider the $\mathbb{F}_p$-algebra $V$ generated by $ a,b,c$ and the relations
$$u a^2 = bc + cb$$
$$v b^2 = ac + ca$$
$$w c^2 = ab + ba$$
Is $V$ generated by ...
- 61.5k
9
votes
2
answers
642
views
When are two semidirect products of two cyclic groups isomorphic
(I have posted this question in Math Stack Exchange, only to have received no answer.)
It is well known that a semidirect product of two cyclic groups $C_m$ and $C_n$ has the form
$$
C_m \rtimes_k C_n ...
- 465
5
votes
1
answer
201
views
Equivalences of categories of complexes of modules
Let $k$ be an algebraically closed field of characteristic $0$.
Let $R, S$ be two commutative $k$-algebras.
Let $\operatorname{Ch}(R), \operatorname{Ch}(S)$ be the categories of complexes of $R$-...
- 639
6
votes
0
answers
236
views
Torsion in the Lie algebra cohomology of gl(n,Z)
What is known about the Lie algebra cohomology $H^*(\mathfrak{gl}_n(\mathbb{Z}),\mathbb{Z})$? After passing to $\mathbb{Q}$-coefficients, the question is classical: $H^*(\mathfrak{gl}_n(\mathbb{Q}),\...
- 4,467
0
votes
0
answers
42
views
Reference request for the smallest set( resp. abelian group) that a group ( resp. ring ) has a faithful action on
For a proof of the Cayley's Theorem, it is obvious to see that a group ( resp. ring ) has a faithful action on itself by left-multiplication. I would like to extend the result for a bit and find the ...
- 101
2
votes
1
answer
266
views
Is there a combinatorial interpretation for the change of basis matrix in the Frobenius normal form representation?
Let $G$ be a graph on $n$ vertices. Let $A$ be the adjacency matrix of $G$ (i.e., rows and columns of $A$ are indexed by vertices of $G$, and the $(v,w)$ entry of $A$ is $1$ if $(v,w)$ is an edge in $...
- 455
5
votes
1
answer
160
views
Relation between row space and column space resp. null space and left null space over general rings
Let $R$ be a ring and $M\in\text{Mat}(R,m\times n)$ a matrix for $m,n\in\mathbb{N}$. What results are known about the relation between column space (cs, image) and row space (rs), resp. null space (...
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9
votes
0
answers
417
views
Does Wedderburn's Theorem hold constructively?
Wedderburn's Theorem states that every finite division ring is commutative. Perhaps even more surprising, this implies that every finite reduced ring is commutative. The proofs that I am aware of ...
- 61.5k
1
vote
1
answer
122
views
A non-example of a graded Frobenius algebra
Take the class of finite dimensional graded algebras $A = \sum_i A_i$ satisfying $|A_n| = 1$ where $A_n \neq 0$ and $A_m = 0$, for all $m > n$. What is an example in this class that is not ...
10
votes
0
answers
478
views
Reconstruction of commutative differential graded algebras
Let $k$ be an algebraically closed field of characteristic $0$.
Let $A,B$ be commutative differential graded algebras (cdga) over $k$ such that $H^{i}(A)=H^{i}(B) =0 \ (i>0)$.
Here, differentials ...
- 639
0
votes
0
answers
58
views
Prime index subgroups of $\langle Q^{i}(\mathbb Z^{2}) \mid i \in \mathbb Z \rangle$ that is invariant under matrix $Q$
Let $Q $ be a matrix in $ \operatorname{GL}(2, \mathbb{Q}) $ and consider the group
$G = \langle Q^{i}(\mathbb Z^{2}) \mid i \in \mathbb Z \rangle := \langle Q^{i}(v) \mid i \in \mathbb Z, v \in \...
- 763
3
votes
1
answer
363
views
Subalgebras of quadratic algebras that are not quadratic
Suppose $A=k\oplus A_1 \oplus A_2\oplus \cdots$ is a quadratic algebra over a field $k$. Let $B$ be the subalgebra generated by a subspace $V\subseteq A_1$. What are the examples of such subalgebras $...
0
votes
0
answers
62
views
Continuity of linear map on tensor product spaces with different norm properties
I originally asked this question on StackExchange, but I think that it may be more suitable to here.
Let $V$ and $U$ be Banach spaces. I'm considering a linear map $\phi: V \rightarrow U$, and ...
- 177
7
votes
1
answer
598
views
Given a rational matrix $Q$, can we generate $\langle Q^{i}(v)\mid i\in\mathbb Z,v\in\mathbb Z^{2}\rangle$ using only non-negative powers of a matrix?
I have copied this question from StackExchange, thank you to those who helped me to improve the question. (apology if you have seen this question already)
Let $Q $ be a matrix in $ \operatorname{GL}(...
- 763
4
votes
1
answer
176
views
Subfields of division rings of degree $2$ which are not invariant
Let $A$ be a noncommutative division ring, and let $B$ be a sub division ring (here, $B$ is allowed to be commutative) of degree $2$. Are there easy examples known for which $B$ is not globally fixed ...
- 4,353
2
votes
1
answer
182
views
Markov property for groups?
My question again refers to the following article:
Koji Fujiwara, Zlil Sela, The rates of growth in a hyperbolic group, Invent. math. 233 (2023) pp 1427–1470, doi:10.1007/s00222-023-01200-w, arXiv:...
- 309
19
votes
2
answers
815
views
The discriminant of the Okada algebra
The Okada algebra $\mathfrak{O}_n$ over a field $K$ has generators
$E_1,\dots,E_{n-1}$ and relations $E_i^2=x_iE_i$,
$E_{i+1}E_iE_{i+1}=y_i E_{i+1}$, and $E_iE_j=E_jE_i$ for $|i-j|\geq
2$, where $x_i,...
- 49.5k
2
votes
0
answers
285
views
A question on Giles Gardam counter example to the Unit conjecture of Kaplansky
The unit version of the Kaplansky conjecture is about units in $FG$ where $F$ is a field and $G$ is a torsion free group. In a recent counter example by Giles Gardam, it is given an ...
- 312
2
votes
0
answers
80
views
Number of minimal generators for a the group algebra of a p-group
Let $G$ be a finite $p$-group and $K$ a field of characteristic $p$.
Then the group algebra $KG$ is local and thus the quotient of a non-commutative polynomial ring $K\langle x_i\rangle$ by an ...
- 26.1k
8
votes
2
answers
609
views
Analogous results in geometric group theory and Riemannian geometry?
As you can see from my other question I concern mmyself with the following article at the moment:
Koji Fujiwara, Zlil Sela, The rates of growth in a hyperbolic group, Invent. math. 233 (2023) pp 1427–...
- 309
6
votes
1
answer
351
views
Morita equivalences and centers of some algebras
Let $k $ is an algebraically closed field of $\text{ch}(k)=0$.
Let $$E := k \langle x_0, x_1, x_2 ,x_3 \rangle/(x_ix_j-q_{ij}x_jx_i )_{0 \leq i,j \leq 3},$$ where $$(\text{deg}(x_0), \text{deg}(x_1), \...
- 639
9
votes
1
answer
704
views
Where has this structure been observed?
$\newcommand{\M}{\mathcal{M}}$Let $M$ be a monoid. Consider the following structure:
$R_X,R_Y:\mathbb{Z}^2 \to M$ satisfying the following "compatiblity-relation":
$$R_X (x, y) \cdot R_Y (x +...
- 6,621