All Questions
Tagged with rings-and-algebras or ra.rings-and-algebras
3,500 questions
-2
votes
0
answers
95
views
A presentation for the group $GL(n,\mathbb{Z}_p)$
Let $n\ge 2$. Let $p$ be a prime and $\mathbb{Z}_p$ denote the finite field with $p$ elements.
I want to know about the presentation for the group $GL(n,\mathbb{Z}_p)$ consisting of its generators and ...
- 105
0
votes
0
answers
45
views
Artinian simple algebras with involution
Let $A$ be an Artinian simple $K$-algebra with involution $*$. The algebra $A$ has a primitive idempotent $e$, and $D_e:=eAe$ is a division $K$-algebra due to Schur's lemma and the isomorphism $(eAe)^{...
- 143
2
votes
0
answers
126
views
A property of matrices formed by pairing of roots and coroots
Let $A$ be an $n\times n$ integral matrix, define its level $l(A)$ as
$$l(A) := \begin{cases}0 &\det A = 0 \\ \text{smallest integer } N \text{ such that } NA^{-1} \text{ is integral} &\det A \...
- 528
2
votes
1
answer
152
views
On generation of $A_n$ by elements of prime order
There is a question regarding generation of finite simple groups with elements of prime order. Recently, Guralnick, Shareshian, Woodroofe and Teräväinen made advances in this direction. We have, for ...
- 329
2
votes
1
answer
143
views
Baer sums of extensions
Apologies in advance if this question is too basic, I looked briefly through Weibel and the stacks project and couldn't find any relevant reference.
Let $\mathcal{A}$ denote an abelian category, and ...
- 2,907
8
votes
1
answer
435
views
Function $\phi$ such that $f(\phi(x,y)) = f(x) + f(y)$
I have a continuous function $f:\mathbb{R}^n\to\mathbb{R}$, and I am looking for a continuous (or at least measurable) function $\phi:\mathbb{R}^{2n}\to\mathbb{R}^n$ such that $f(\phi(x,y))=f(x)+f(y)$....
- 3,065
1
vote
1
answer
79
views
Hilbert symbol of a quaternion algebra given ramified places
I am reading the paper: https://projecteuclid.org/journals/experimental-mathematics/volume-17/issue-3/Derived-Arithmetic-Fuchsian-Groups-of-Genus-Two/em/1227121388.full
in order to find an explicit ...
- 155
3
votes
1
answer
113
views
Selmer complex and total complex
Thanks for your reading. I'm studying Selmer complex book by Jan Nekovar. For the definition of Selmer complex I meet a problem.
In the introduction(page 9, 0.8.0) the author gives us a definition of ...
- 319
13
votes
2
answers
1k
views
What's the deal with De Morgan algebras and Kleene algebras?
The notion of Boolean algebras, and the corresponding classical propositional logic, is very standard, and it is easy to find information about them (for example, among many other such works, there is ...
- 32.4k
6
votes
0
answers
101
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Computer program for free restricted Lie polynomial
I am conducting numerical experiments involving the Gröbner–Shirshov Basis for restricted Lie algebras. At each step of the computation, I need to work with restricted Lie polynomials. Specifically, I ...
- 1,013
3
votes
1
answer
103
views
References: rigorous algorithms for elementary computations in base-b with complexity estimates
Definitions/Notation: Fix positive integers $b$ and $M$. Consider the set of real numbers which can be exactly expressed with $2M+1$ coefficients in base $b$, defined by
$$\mathcal{X}(b,M):=\{x\in \...
- 5,407
47
votes
10
answers
6k
views
Algebraic theorems with no known algebraic proofs
What are some good examples of algebraic theorems that have no known algebraic proofs?
A few I know concern classifications of (not necessarily associative) division algebras over $\mathbb{R}$: the ...
2
votes
0
answers
93
views
Free, easy-to-use program for noncommutative algebra over finite fields
I am looking for a computer program that can handle computations in noncommutative algebra over a finite field of prime order $p$.
My requirements are:
The program should be free, as I do not have ...
- 1,013
4
votes
1
answer
239
views
Connected Frobenius algebras non-semisimple as an object
A Frobenius algebra object $A$ in a tensor category $\mathcal C$ is said to be connected if $\text{Hom}_{\mathcal C}(\mathbb{1}, A)$ is a one dimensional vector space, where $\mathbb {1} $ denotes the ...
- 43
4
votes
2
answers
365
views
Notion of prime congruences
We have the idea of a prime ideal in a commutative ring $R$ but in universal algebra, we generalize the notion of ideal to that of a congruence. I’ve thought over the question of what a prime ...
- 293
1
vote
0
answers
152
views
Does it make sense to take a formal power series over a non-ring? [closed]
Suppose we have the set of nonnegative integers. This is obviously not a ring, and not even a group. But are we able to identify ring-like properties for it if we take a formal power series over it?
...
- 29
0
votes
0
answers
31
views
Right maximal ideals in skew-Laurent rings over division Rings
Let $R$ be a Noetherian domain, and let $D$ denote its division ring. Define $S = D_q[x_1^{\pm 1}, x_2^{\pm 1}]$ as an iterated skew-Laurent polynomial ring with the relation $x_1 x_2 = q x_2 x_1$. Is ...
- 923
0
votes
0
answers
91
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
0
votes
0
answers
61
views
$\mathcal{R}$ is finite over $L_0[e,A]$
Let $\sigma:L \longrightarrow L$ an automorphism of infinite order, where $L_0 \subset L$ is its fixed field. Let $R$ be a commutative subring of $L\{\sigma\}$. Let
\begin{equation}
\mathcal{R}=\...
- 69
4
votes
0
answers
131
views
Ring theoretical aspects of the DAHA
The double affine Hecke algebras (DAHA) were introduced by Cherednik in his study of Macdonald's inner product conjectures (which were solved affirmatively).
Nowdays there are many variations of the ...
- 3,318
6
votes
0
answers
138
views
Can an algebra be isomorphic to its own algebra of $n^2 \times n^2$ matrices but not its own algebra of $n \times n$ matrices?
Is there an associative unital algebra $A$ which is isomorphic to its own algebra of $n^2\times n^2$ matrices $\operatorname{Mat}_{n^2}(A)$, but not isomorphic to its algebra of $n \times n$ matrices $...
- 54.6k
2
votes
0
answers
76
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
0
answers
69
views
Link between Carathéodory's criterion and commutation in an orthomodular lattice?
In the theory of outer measures, Carathéodory's criterion constructs from an outer measure $\mu^*$ on $X$ a $\sigma$-algebra $\Sigma$ of subsets of $X$, on which the restriction of $\mu^*$ is a ...
- 193
9
votes
3
answers
1k
views
Examples of combinatorial problems where the only known solutions, or most "natural" solutions, use representation theory?
In Solution of two difficult combinatorial problems with linear algebra, Robert Proctor presents two simply stated combinatorial problems, and gives solutions to them using a linear algebraic approach ...
4
votes
0
answers
211
views
When does a short exact sequence of abelian groups with $B\cong A\oplus C$ split?
$\hspace{20pt}$Duplicate on stackexchange.
This question, in a way, extends this one. The question is what are some sufficient conditions on the abelian group $B$ so that if $B\cong A\oplus C$ and a ...
- 171
4
votes
1
answer
347
views
Do Frobenius subalgebras form a lattice?
A finite-dimensional, unital, associative algebra $A$ over a field $k$ is termed a Frobenius algebra if it is endowed with a nondegenerate bilinear form $\sigma : A \times A \to k$ satisfying the ...
5
votes
1
answer
178
views
Semisimplicity of algebras in fusion categories
Let $\mathcal{C}$ be a fusion category and $A \in \mathcal{C}$ be an algebra object. We say that $A$ is semisimple if its category of (right) modules $\mathsf{mod}_A(\mathcal{C})$ is a semisimple ...
- 61
5
votes
0
answers
112
views
Finitely generated projective modules over Noetherian endomorphism ring
Let $\mathcal A$ be a locally Noetherian Grothendieck abelian category and $M\in \mathcal A$ be a Noetherian object. Set $B:=\text{End}_{\mathcal A}(M)$. Let $B$-mod be the category of finitely ...
- 413
0
votes
0
answers
54
views
What properties do these "norm-equal" polynomials have?
Let us first define the "norm-equal" polynomials :
For $f(x),g(x)\in \mathbb{C}[x]$, if $\forall z\in \mathbb{C},|z|=1$, we have $|f(z)|=|g(z)|$, then we call $f(x)$ and $g(x)$ are "...
- 19
1
vote
0
answers
92
views
Proposition 6.2.7 from Goss
I'm following David Goss's book Basic structures of function field arithmetic. Let $L$ be an extension field of $L_0$ and $\sigma$ an automorphism of infinite order which fixes $L_0$. Let $L\{\sigma\}$...
- 69
0
votes
0
answers
104
views
Non-degenerate bilinear pairing of finite dimensional algebras
A finite dimensional algebra (over $\mathbb{C}$, say) is said to be Frobenius if it comes equipped with a nondegenerate bilinear pairing
\begin{align*}
\langle -, - \rangle : A \times A \rightarrow \...
- 71
2
votes
1
answer
198
views
Maximal sub-$\mathbb{C}$-algebras of $\mathbb{C}[x,y]$
After asking this question and finding this relevant paper, I would like to ask the following question:
For every $a,b \in \mathbb{C}$, denote:
$A_{a,b}=\mathbb{C}[(x-a)(x-b),x(x-a)(x-b),y]$
and
$B_{a,...
- 2,837
6
votes
3
answers
490
views
Boolean rings with many automorphisms
Does there exist an infinite Boolean ring $R$ (not assume unital, only associative) with the property that for any nonzero $x,y\in R$, there is a ring automorphism $\varphi\colon R\to R$ such that $\...
- 131
4
votes
1
answer
239
views
True or false? Every left or right cancellative, duo semigroup is cancellative
A semigroup $S$ is duo if $aS = Sa$ for all $a \in S$, where $aS := \{ax: x \in S\}$ and similarly for $Sa$; for instance, every commutative semigroup is duo, and so is every group. On the other hand, ...
- 10.5k
5
votes
1
answer
883
views
Is this ring isomorphic to a quotient of a group algebra?
Consider the quotient of the free algebra $\mathbb{Q}\langle \alpha, \beta, \gamma, \delta, \varepsilon, \zeta \rangle$ by the two-sided ideal $I$ subject to the relations $$ \alpha\delta=\delta\alpha=...
- 1,093
1
vote
0
answers
60
views
Reference for Gröbner-Shirshov algorithm in free restricted Lie algebras
I am searching for a reference on the Gröbner-Shirshov algorithm specifically for free restricted Lie algebras. I have already consulted the textbook by Bokut et al (Gröbner–Shirshov Bases Normal ...
- 1,013
8
votes
1
answer
320
views
Does every cancellative duo semigroup embed into a group?
Prompted by the comments to a recent answer by YCor to a related question (here), I'd like to ask the following:
Q. Does every cancellative duo semigroup embed into a group?
A (multiplicatively ...
- 10.5k
1
vote
0
answers
89
views
The base group of a wreath product of an abelian group by $ {\mathbb{Z}}$ is a characterstic subgroup
I've copied over this question from what I asked on Mathematics Stack Exchange, in the hope that some experts here can direct me to some relevant results.
Let $A$ be a finitely generated abelian group,...
- 823
8
votes
2
answers
596
views
If a semigroup embeds into a group, then is it a subdirect product of groups?
The title has it all:
Q. If a semigroup $S$ embeds into a group, then is $S$ (isomorphic to) a subdirect product of groups?
If yes, then $S$ is a subdirect product of subdirectly irreducible groups,...
- 10.5k
5
votes
0
answers
216
views
Lifting a morphism between quasi-projective varieties
Let $\mathcal{V}$ be an affine algebraic variety over $\mathbb{R}$, $G$ be a finite group acting freely on $\mathcal{V}$. Consider the quotient space $Y:=\mathcal{V}/G$, which itself is a quasi-...
- 364
15
votes
1
answer
518
views
Pairs of matrices for which traces of powers are independent of the order
Let $A,B$ be $n\times n$ matrices over ${\mathbb C}$ such that, for all $m,k$ and all partitions $(i_1,\ldots ,i_r)$ of $m$ and $(j_1,\ldots ,j_r)$ of $k$ (perhaps with some zero parts),
$${\rm tr}\, (...
- 1,336
7
votes
2
answers
487
views
Is every cancellative semigroup a subdirect product of subdirectly irreducible cancellative semigroups?
By a classical result of Birkhoff (that is, Theorem 2 in [G. Birkhoff, Subdirect unions in universal algebra, Bull. AMS, 1944]) and the trivial fact that the class of semigroups is closed under the ...
- 10.5k
4
votes
0
answers
267
views
If $\mathbb{C}[a,b,c] \subsetneq \mathbb{C}[x]$, then there exist $f,g$ s.t. $\mathbb{C}[a,b,c] \subseteq \mathbb{C}[f,g] \subsetneq \mathbb{C}[x]$
I ran into this MSE question and would like to ask about its answer and plausible generalizations.
The quoted MSE question asks if the following claim is true or false and why:
Claim: Let $a,b,c \in \...
- 2,837
0
votes
1
answer
222
views
On the Irreducibility of Cyclotomic polynomials
Let $F$ be any field with $\operatorname{Char} F=q$. Let $p$ be a prime such that $p\neq q$. Suppose $F$ has no $p$-th root of unity except $1$. Is it true that the cyclotomic polynomial $X^{p-1}+\...
- 494
2
votes
1
answer
327
views
Completion of a local ring is noetherian (under some hypothesis)
I was reading the proof of Lemma 10.12 in this paper. In the second sentence, the following fact is used implicitly:
Let $(R,\mathfrak{m})$ be a commutative local ring. Let $\widehat{R}$ be its $\...
- 293
8
votes
1
answer
361
views
Invertible matrix with group ring coefficient
Before asking the question I do need
some notations.
$G$ a (torsion-free) group, $\mathbb{Z}^{´}=\mathbb{Z}[\frac{1}{2}]$
$R:= \mathbb{Z}[G]$, $R^{´}=\mathbb{Z}^{´}[G]$ group rings.
$Mat_{n}(R)$ the ...
- 223
2
votes
0
answers
73
views
Are idempotents in a nonnegatively graded algebra conjugate to homogeneous idempotents?
Let $k$ be a field (assume algebraically closed and $\operatorname{char}(k) = 0$ if it helps) and $R = \bigoplus_{i \geq 0} R_i$ a (unital associative) non-negatively graded $k$-algebra. Furthermore, ...
5
votes
0
answers
226
views
Cohomology of representation varieties and the Hochschild cohomology
Let $k$ be a field, $A$ a $k$-algebra, and $V$ a $k$-vector space. Then we can consider the representation varieties of $A$ on $V$: $\mathrm{Hom}_{k\textrm{-alg}}(A, \mathfrak{gl}(V))$ and $\mathrm{...
- 985
3
votes
1
answer
108
views
Roots of polynomial $\sum_{\sigma \in W} x^{l(\sigma)}$
Let $W$ be Weyl group of a root system $\Phi$ (of finite dimensional simple Lie algebra). For $\sigma\in W$, $l(\sigma)$ be the its length. Consider the following polynomial
$$P_\Phi(x) = \sum_{\sigma ...
- 528
6
votes
1
answer
315
views
Sum of derivative of polynomial over its simple roots
Let $P$ and $Q$ be polynomials over $\mathbb C$, and $n\in\mathbb N$ be a positive integer. I'm interested in the root sums of the form
$$ \sum_{P(x)=0}\frac{Q(x)}{P'(x)^n},$$
where the sum runs over ...
- 99