All Questions
1,123 questions
6
votes
0
answers
190
views
The highest degree of a polynomial on a finite group
This question is motivated by the comments and the answer to this MO-question.
First let us recall some definitions.
A function $f:X\to X$ on a group $X$ is called a polynomial if there exists $n\in\...
- 41.9k
0
votes
0
answers
41
views
Polyextremal groups
A polynomial of a semigroup $X$ is a function $f:X\to X$ of the form
$f(x)=a_0xa_1\cdots xa_n$, where $a_0,a_1,\dots,a_n$ some elements of the semigroup $X^1=X\cup\{1\}$, called the coefficients of ...
- 41.9k
1
vote
0
answers
53
views
The "hyperbolicity preserving" probabilities
A classical fact (due to Polya ?) is that if $P\in{\mathbb R}[X]$ has only real roots (one says that $P$ is hyperbolic), and $a$ is a real number, then the roots of
$$L_aP(X):=\frac12(P(X+ia) +P(X-ia))...
- 52.3k
6
votes
1
answer
319
views
Does there exist any massive proper $C^*$-subalgebra?
Definition 1: Suppose $B$ is a $C^* $-algebra. $A$ is massive $C^* $-subalgebra of $B$ iff
1. $A$ is a subalgebra of $B$;
2. for each irreducible representation $\pi$ of $B$ representation $\pi|_A$ is ...
- 1,446
4
votes
1
answer
265
views
The going-up theorem for free extensions of almost commutative rings
I would like to know whether or not the going-up property holds for some classes of finite filtered extensions of non-commutative rings.
Let $S \subseteq R$ be rings. The pair $(S,R)$ has the going-...
CommunityBot
- 1
2
votes
0
answers
181
views
So many types of subwords! How are they called?
Let $\mathscr F(X)$ be the free monoid on an alphabet $X$, the carrier set of $\mathscr F(X)$ being the union of $X^{\times k}$ (the Cartesian product of $k$ copies of $X$) as $k$ ranges over $\mathbb ...
- 10.5k
0
votes
0
answers
62
views
To find a DFT for complex functions on a semigroup
For a certain commutative semigroup of integer size $n$, $G=(\{1,2,\dots,n\},\circ: x\circ y\mapsto \min(n,x+y))$, consider all complex functions on it, denoted by $\mathbb C[G]$ or $\mathbb CG$. ...
7
votes
1
answer
179
views
Symmetry of unique generator property
In this article:
Canfell, M. J. "Completion of Diagrams by Automorphisms and Bass′ First Stable Range Condition." Journal of algebra 176.2 (1995): 480-503.
the author defines a ring $R$ to ...
- 1,507
12
votes
2
answers
914
views
non commutative polynomial which is zero for all matrix evaluation
Let $K$ be a (commutative) field.
We can define the free $K$-algebra of polynomials in non commutative variables $x_1, \cdots, x_n$. It is usually denoted by $K\langle x_1, \cdots, x_n \rangle$.
Fix a ...
- 33.8k
12
votes
1
answer
624
views
Stone–Čech compactification as a semigroup
Let $G$ be a topological group (we can assume that $G$ is countable and discrete) and let $\beta(G)$ be the Stone–Čech compactification of $G$. It is known that $\beta(G)$ can be turned into a left ...
- 38.6k
3
votes
0
answers
31
views
Compactness of the minimal ideal of a compact Hausdorff polytopological semigroup
A semigroup $X$ endowed with a topology is called
$\bullet$ a topological semigroup if the semigroup operation $X\times X\to X$ is continuous;
$\bullet$ a semitopological semigroup if for every $a,b\...
- 41.9k
0
votes
0
answers
58
views
An action on multiplicatively antisymmetric matrix
A matrix $ Q=(q_{ij})$ is called multiplicatively antisymmetric over a field $ F $ if $ q_{ii}=1 $ and $ q_{ij}={q_{ji}}^{-1} $.Let $ \mathcal{Q} $ be the set of all $ n \times n $ multiplicatively ...
- 923
7
votes
0
answers
138
views
The smallest cardinality of a cover of a group by algebraic sets
$\DeclareMathOperator\cov{cov}$A subset $A$ of a semigroup $X$ is called algebraic if $$A=\{x\in X: a_0xa_1x...xa_n=b\}$$ for some $b\in X$ and $a_0,a_1,...,a_n \in X^1=X\cup \{1\}$. The smallest ...
- 41.9k
14
votes
1
answer
545
views
Is the discriminant of a free (as a module) $R$-algebra always congruent to a square modulo 4?
Let $R$ be a commutative ring. Let $A$ be an $R$-algebra (i.e., an $R$-module
equipped with an $R$-bilinear multiplication map that turns $A$ into a unital
ring). We do not require $A$ to be ...
- 33.8k
42
votes
5
answers
4k
views
What are the main structure theorems on finitely generated commutative monoids?
I should read J. C. Rosales and P. A. García-Sánchez's book Finitely Generated Commutative Monoids and L. Redei's book The Theory of Finitely Generated Commutative Semigroups. I haven't. But here's ...
- 10.5k
2
votes
0
answers
172
views
Simple modules of quantum planes
Let $k$ be an algebraically closed field.
Let $R := k\langle x,y \rangle/(yx-qxy) (q \in k^*)$.
We often call $R$ a quantum plane.
If $q$ is a primitive $n$-th root, then for any $(\zeta, \xi) \in k^* ...
- 889
6
votes
1
answer
135
views
Automorphisms of special egg-box diagrams
By a egg-box diagram I will simply mean a (possibly infinite) rectangular array of holes, with some of the holes containing an egg (denoted by a filled-in circle) and the rest of the holes are empty (...
- 18.7k
3
votes
2
answers
172
views
On the Hilbert function of a numerical semigroup
Recall that a numerical semigroup $S$ is a submonoid of the non-negative integers $\mathbb Z_{\geq 0}$ whose relative complement $\mathbb Z_{\geq 0} \setminus S$ is finite. Observe that the collection ...
- 183
2
votes
1
answer
94
views
What are the n-ary subsemigroups of $\mathbb{N}$?
There is a well-known result about the subsemigroups of $\mathbb{N}$ stating that the additive subsemigroup generated by a (finite) set $A$ of $\mathbb{N}$ is cofinite in $\mathbb{N}$ if and only if $\...
- 63.9k
21
votes
1
answer
2k
views
Is there any non-commutative ring such that every element other than the identity is a zero divisor?
A (unital) ring $R$ with the property that every element other than the identity $1_R$ is a (two-sided) zero divisor, seems to be commonly called a "$0$-ring" or "$\mathcal O$-ring"...
- 495
0
votes
0
answers
39
views
Countably infinite monoids with minimal right ideals
Is there any classification of countably infinite monoids with minimal right ideal? or at least in some classes of monoids?
- 237
2
votes
0
answers
88
views
The generators of twisted homogeneous coordinate rings
Let $X$ be a projective scheme over an algebraically closed field $k$ of characteristic $0$.
Let $\sigma$ is an automorphism of $X$ and $\mathcal{L}$ be an invertible sheaf on $X$.
Let $B := B(X, \...
- 889
41
votes
4
answers
2k
views
What is the probability two random maps on n symbols commute?
It is well known that two randomly chosen permutations of $n$ symbols commute with probability $p_n/n!$ where $p_n$ is the number of partitions of $n$. This is a special case of the fact that in a ...
- 7,545
7
votes
1
answer
284
views
Does the choice of the algebraically closed field of characteristic $p$ have influence on the module category?
Let $G$ be a finite group and $p$ be a prime number dividing $|G|$.
Let $k$ be the algebraic closure of $\mathbb{F}_p$.
Let $K$ be another algebraically closed field of characteristic $p$ which is not ...
- 38.6k
7
votes
1
answer
271
views
Algebraic proof that the monoid ring of a torsion-free monoid is reduced
In what follows, I say that a monoid $M$ is torsion-free if the $n$-th power map is injective for all $n \geq 1$. I have a proof of the following result:
Claim: if $M$ is a torsion-free commutative ...
- 15.5k
2
votes
2
answers
134
views
On a generating set of numerical semigroups of multiplicity three
Let $S$ be a numerical semigroup. Let $\mathbb N$ denote the monoid of non-negative integers under addition. Let $F(S)=\max (\mathbb N \setminus S)$ be the Frobenius number of $S$; let $g(S)=|\mathbb ...
- 785
5
votes
1
answer
330
views
About a recent paper of Rickard on finitistic dimension
Apologizes if this is a basic question, but I am new to the area of finite dimensional algebras. I am reading the paper "Unbounded derived categories and the finitistic dimension conjecture" ...
- 30.3k
3
votes
0
answers
109
views
Noncommutative group schemes corresponding to quantum groups
I'm not an expert on quantum groups by any stretch, so forgive me if this question seems overly naive. That said, I was wondering if there is a way (or if there has been any attempt in the literature) ...
- 1,516
2
votes
0
answers
101
views
What is the relationship between ramification in central simple algebras and in fields?
Suppose $K$ is the field of fractions of a Dedekind domain $R$, and let $L$ be a finite extension of $K$. There is a notion of ramification of primes of $K$ in $L$, which describes how $\mathfrak p \...
- 121
17
votes
2
answers
2k
views
How much theory works out for "almost commutative" rings?
I've been reading about D-modules, and have seen a proof that D_X, the ring of differential operators on a variety, is "almost commutative", that is, that its associated graded ring is commutative. ...
- 1,446
3
votes
0
answers
174
views
On continuous seminorms on Fréchet-Stein algebras
Let $K$ be a discretely valued complete non-archimedean field and $U$ be a left Fréchet-Stein algebra as defined in Algebras of p-adic distributions and admissible representations, with a Fréchet-...
- 541
6
votes
0
answers
255
views
Proving the spectrum of the Young-Jucys-Murphy elements by formal computation in the degenerate affine Hecke algebra
This is really a followup to Why are Jucys-Murphy elements' eigenvalues whole numbers? , specifically to Igor Makhlin's beautiful answer. I'm trying to make it even more beautiful by getting rid ...
- 33.8k
15
votes
2
answers
655
views
Indecomposable contracting maps on the integers
$\def\ZZ{\mathbb{Z}}$Call a function $f : \ZZ \to \ZZ$ "contracting" if
$$|f(j) - f(i)| \leq |j-i|$$
for all $i$, $j \in \ZZ$. The contracting functions form a monoid under composition; call ...
- 148k
2
votes
1
answer
287
views
On the definition of the Cherednik algebra of a variety with a finite group action
Let $X$ be a connected complex smooth affine variety, acted on by a finite group $G$. We define a reflection hypersurface $(Y,g)$ as a smooth codimension one subvariety $Y\subset X$ which is fixed by $...
- 516
0
votes
1
answer
72
views
Reference request: Left $R/k$-modules [closed]
In the paper titled:
On the module of differentials of a noncommutative algebra and symmetric biderivations of a semiprime algebra
I found the following definition:
Let $k$ be a commutative ring with ...
- 5,878
3
votes
1
answer
137
views
An f.g.u. duo monoid is unit-duo: True or false?
Let $H$ be a monoid (written multiplicatively) with the property that $H = H^\times A H^\times$ for some finite $A \subseteq H$ (shortly, an f.g.u. monoid), where $H^\times$ is the group of units of $...
- 38.6k
13
votes
1
answer
623
views
Ultracategories with one object
Historically, the theory of ultracategories was invented by Makkai to prove a strong conceptual completeness theorem for first-order logic, roughly: if $T$ and $S$ are two first-order theories such ...
- 15.8k
11
votes
3
answers
1k
views
Existence of non-commutative desingularizations
Let $R$ be normal, local ring of dimension at least $2$. Let $M$ be a reflexive $R$-module and let $A=Hom_R(M,M)$. Suppose $A$ has finite global dimension. Then one can view $A$ as a weak non-...
- 4,713
2
votes
0
answers
67
views
Type of numerical semigroups is not bounded when embedding dimension is $\geq 4$
I am currently studying numerical semigroups. I know that there is no upper bound for the type of a numerical semigroup with embedding dimension greater or equal than $4$. There is a famous example by ...
- 121
14
votes
1
answer
2k
views
When are epimorphisms of algebraic objects surjective?
Let $C$ be the category of $\tau$-algebras for some type $\tau$. Consider the statements:
Every monomorphism is regular.
Every epimorphism in $C$ is surjective.
It is easy to see that 1. implies 2. ...
- 1,446
3
votes
1
answer
409
views
When are simple holonomic D-modules of the form $\mathcal{D}/\mathcal{D}L$?
Let $\mathcal{D}=\mathcal{D}_X$ be the sheaf of rings of differential operators on a smooth algebraic curve $X$.
Since $\dim X=1$, the D-modules of the form $\mathcal{D}/\mathcal{D}L$ are necessarily ...
- 711
5
votes
0
answers
138
views
Can we define partial group actions on (finite) sets via generators and relators?
Let $G = \langle Y | R \rangle$ be a finitely presented group. A partial group action on a set $X$ is a premorphism into the inverse semigroup
$$
\mathcal I (X) = \{ f: A \to B : A, B \subseteq X, f\...
- 639
2
votes
1
answer
246
views
The combinatorics of $(f \partial)^n$ in the noncommutative setting?
This is a noncommutative version of these three previous questions:
differential operator power coefficients
Сlosed formula for $(g\partial)^n$
A Leibniz-like formula for $(f(x) \frac{d}{dx})^n f(x)$?
...
- 12.9k
5
votes
1
answer
300
views
Quasi-coherent cohomology in non-commutative algebraic geometry
In non-commutative algebraic geometry, the motto so to speak is to replace the study of a scheme $X$ with the study of the category $D_{qcoh}(X)$ of quasi-coherent sheaves and study the properties ...
- 39.3k
3
votes
0
answers
317
views
How to go from primitive idempotents in $\text{End}_A(M)$ to primitive idempotents in $A$?
Let $K$ be a large enough finite field, let $A$ be a finite-dimensional $K$-algebra.
Moreover, let $M$ be a finitely generated $A$-module and let $M = M_1\oplus ... \oplus M_n$ be a decomposition of $...
- 1,755
4
votes
0
answers
91
views
Nullstellensatz for maximal left ideals of quantum plane
Let $R=\mathbb{C}\langle x,y\rangle/\langle xy=qyx\rangle$ be the quantum plane algebra. Does some sort of Nullstellensatz holds for the maximal left ideals of $R$? By this we mean all maximal left ...
- 91
13
votes
5
answers
3k
views
Noncommutative localization of a ring: complete construction
I've been looking for the following construction in the literature, but I've only been able to find (very) partial proofs or proofs of special cases.
Let $R$ be a non-commutative ring and $S$ a ...
- 63.9k
19
votes
1
answer
1k
views
When should I expect a quiver with potential to be rigid?
This question is pretty technical, but there are some very smart people here.
Fix a quiver Q, WITH oriented cycles. Let k[[Q]] be the completed path algebra. (Like the path algebra, but we allow ...
- 35.5k
29
votes
1
answer
1k
views
Is the Golomb countable connected space topologically rigid?
The Golomb space $\mathbb G$ is the set of positive integers endowed with the topology generated by the base consisting of the arithmetic progressions $a+b\mathbb N_0$ with relatively prime $a,b$ and $...
- 63.9k
5
votes
1
answer
273
views
'Lie correspondence' for formal power series in non-commuting indeterminates
This is related to an earlier question of mine. I would like an argument or a reference (or a missing hypothesis if needed) for the following.
Let $\mathbb{F}\langle\langle \alpha\rangle\rangle$ and ...
- 2,388