All Questions
1,123 questions
6
votes
1
answer
335
views
A potential resolution of $R/r$
The DGA
For $k$ some field, let $R$ be a $k$-algebra, and let $r\in R$.
Define a differential graded algebra $\mathbf{R}_r$ as follows. As a graded algebra, it is isomorphic to $R\langle t\rangle$, ...
CommunityBot
- 1
7
votes
1
answer
433
views
Powers of maps on finite sets
Let $X_n$ be a set with $n$ elements. Write $F(X_n,X_n)$ for the set of maps from $X_n$ to itself. It is a monoid under the operation of composition. Let $m$ be a positive integer. How many maps in $...
- 13k
4
votes
3
answers
1k
views
Set of invertible operators in B(H) is connected. Is it true? Is there a reference?
Suppose $H$ is a Hilbert space, $B(H)$ is the algebra of bounded linear operators on it, $K(H)$ is ideal of compact operators in $B(H)$, $Inv(B(H)/K(H))$ is the topological group of invertible ...
- 11.1k
2
votes
2
answers
492
views
Model Theoretic Localization
This is a re-post on a previous question I asked. My first question was too vague to warrant detailed responses. Really, I have two specific questions to ask.
1) Let $\sigma = (A; \{0,1\}; +, \times)...
- 73.2k
2
votes
1
answer
259
views
Nuclearity of certain semigroup crossed product C*-algebras
This question is related to this question link.
Suppose we have an (abelian) semigroup $S$ acting by endomorphisms on a $C^*$-algebra A giving rise to a semigroup crossed product $B = A\rtimes S$. ...
CommunityBot
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8
votes
1
answer
1k
views
Give an example of monoid with property $m^2 = m^3$
Give an example of finitely generated, infinite monoid $M$ with property that for all $m \in M$ we've got $m^2 = m^3$.
This question comes from the problem I was given during algebraic languages ...
CommunityBot
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7
votes
1
answer
650
views
Cones, monoids, and the space of (very) ample divisors
An interesting and useful tool to study a projective variety is its ample cone. Understanding the structure of this cone reveals information about the variety, and it is an isomorphism-invariant so ...
- 66.3k
4
votes
0
answers
727
views
What is a "double star-product"
Michel Van den Bergh introduced the notion of a double Poisson algebra. The definition is cooked up such that the representation varieties of such an algebra are Poisson varieties.
Is there a notion ...
- 8,435
2
votes
2
answers
983
views
Torsion in tensor products over noncommutative rings
I know that the problem of torsion in tensor products, even of torsion free modules, is a very delicate thing. Unfortunately i don't have a deeper insight into this subject, so i don't know how things ...
- 3,702
1
vote
0
answers
688
views
The sum of a nilpotent left ideal and a nil left ideal
In class, we recently saw that the sum of 2 two-sided nil ideals is a nil ideal. We were asked to show that the sum of a niplotent left ideal and a nil left ideal is a nil left ideal.
I am having ...
- 33.8k
10
votes
2
answers
752
views
Adding a formal inverse of an element to a free monoid
Let $FM_2=\langle a,b\rangle$ be the free monoid of rank 2. If we add a formal inverse to the word $aba$, we get the free group $F_2$ (because both $a$ and $b$ will have inverses).
Question: For ...
CommunityBot
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1
vote
1
answer
283
views
Ore Extensions and the Construction of the Quantum General Linear Group
In the usual (fomal) construction of the quantum general linear group $GL_q(N)$, an Ore extension is used. See for example Kassel. Why is this necessary? Surely one can just augment the set of ...
- 5,232
0
votes
1
answer
147
views
Small set of acts over a countable monoid?
Given a countable monoid $S$, is the set of all (isomorphic representatives of) $S$-acts a small set?
- 11
1
vote
1
answer
154
views
undecidability in the dynamics of functions $f: \Sigma^* \rightarrow \Sigma^*$
Does anyone know any undecidable problems in the dynamics of functions (not necessarily monoid homomorphisms or anything) from \Sigma^* to \Sigma^* where \Sigma is a finite set? In particular, I'm ...
- 236k
6
votes
2
answers
462
views
need references regarding the elementary theory of free semigroup and free abelian groups
Recently, I read that two free abelian groups $S$ and $T$ have the same elementary theory if and only if rank$S$=rank$T$. Does anyone have a reference with a proof of this? Also, what is known about ...
CommunityBot
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2
votes
1
answer
1k
views
monoid ring and some structure within it - how is it called?
I am amateur - mathematics is my hobby, and I find some strange structure working with toy matrices structure so I try to ask some questions regarding it. Let me allow to introduce some structure ...
- 6,870
15
votes
2
answers
3k
views
when are algebras quiver algebras ?
Good Morning from Belgium,
I'm no stranger to the mantra that quiver-algebras are an extremely powerful tool (see for example the representation theory of finite dimensional algebras). But what is a ...
CommunityBot
- 1
9
votes
2
answers
3k
views
Jacobson radical = intersection of all maximal two-sided ideals
I'm embarassed to ask this question, but the literature on noncommutative rings seems to give this a berth as if it was absolutely trivial and not worth discussing, and I can't prove it, so all I can ...
- 3,702
8
votes
2
answers
2k
views
What is an example of a ring in which the intersection of all maximal two-sided ideals is not equal to the Jacobson radical?
What is an example of a ring in which the intersection of all maximal two-sided ideals is not equal to the Jacobson radical? Wikipedia suggests that any simple ring with a nontrivial right ideal would ...
- 44.7k
6
votes
0
answers
998
views
Generalized Courant-Fischer theorem
Consider some quaternionic matrix $A$. A right eigvenvalue of $A$ is a quaternion $q$ such that $Ax=xq$ for some $x\in \mathbb{H}^n$. Similarly, a left eigenvalue of $A$ is quaternion $q$ such that $...
- 475
2
votes
2
answers
492
views
on existence of matrices X, Y s.t. XAY is diagonal over non-commutative ring
Given $A\in Mat_{n\times n}(R)$ where $R$ is a non-commutative associative ring are there exist any (non-zero) matrices $X, Y\in Mat_{n\times n}(R)$ such that $XAY=diag(a_1, \ldots , a_n)$ for some $...
CommunityBot
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26
votes
1
answer
998
views
Idempotents in Rings of Differential Operators
Differential Operators on General Commutative Rings
Let k be an algebraically closed field of characteristic zero, and let R be a commutative k-algebra. Then a (Grothendieck) differential operator on ...
CommunityBot
- 1
3
votes
1
answer
408
views
Finite Homological Dimension of R/P for all P for module finite non-commutative rings
I have a reasonably precise question which I hope is clear enough to get a nice answer. Let R be a Noetherian non-commutative ring which is finite as a module (and flat/free if it helps) over it's ...
- 318
15
votes
2
answers
2k
views
Why does the Grothendieck group $K_0(R)$ of a ring not depend on our choice of using left modules instead of right modules?
I am under the impression that in the definition of the Grothendieck group $K_0(R)$ of a (non-commutative) ring it doesn't matter whether we apply the usual $K_0$ construction to the exact category of ...
- 22.6k
1
vote
1
answer
400
views
Transitive Semigroups of $2\times 2$ matrices
Suppose $G$ is a semigroup (i.e., closed under matrix multiplication) of invertible $2\times 2$ real matrices. Suppose also that $G$ is transitive i.e., for any two non-zero vectors $u$ and $v$ there ...
- 538
5
votes
2
answers
537
views
If $k[S]$ is noetherian, is S finitely generated?
Let $S$ be a semigroup. If $S$ is abelian, then it follows that the semigroup algebra $k[S]$ is finitely generated if and only if $S$ is.
What if we relax the condition on $k[S]$, so that $k[S]$ is ...
CommunityBot
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8
votes
1
answer
623
views
Is there a good computer package for working with complexes over non-commutative rings?
I'm interested in doing computations with certain non-commutative rings, most of which involve taking derived tensor products. Does anyone know of a computer algebra package which will find ...
0
votes
2
answers
273
views
Identifying a sequence of polynomials
Studying a specific quantum cluster algebra, I have come across the following sequence of polynomials :
$$X_1$$
$$q^{-1/2}(X_1X_2-1)$$
$$q^{-1/2}(X_1X_2X_3-X_3-X_1)$$
$$q^{-1}(X_1X_2X_3X_4-X_3X_4-...
- 44.7k
1
vote
2
answers
378
views
Is this a pre-ordered commutative semigroup?
Motivation
I'm studying an approach to axiomatic thermodynamics based on the notion of commutative semigroup $(S,+)$ with a preorder relation $\to$ on $S$. In other words, $S$ is non-empty set, the ...
- 21
3
votes
0
answers
163
views
Pulling out factors in a Noetherian Domain
Let $R$ be a Noetherian domain (not-necessarily commutative), and let $S$ be a Noetherian subring of $R$. An element $r\in R$ is left $S$-irreducible if, for any $s\in S$ and $r' \in R$ with $sr'=r$, ...
- 13k
2
votes
2
answers
300
views
what conditions can one place on a finitely generated periodic semigroup that will ensure the semigroup is finite?
I am not familiar with much semigroup theory, but this question came up in my research and I've been unable to find much on it.
- 85.6k
5
votes
1
answer
2k
views
Length of a module over different rings
Given a regular local ring $(R,m)$ and a finitely generated $R$-algebra $S$, which is free as an $R$-module. Let $M$ be a left $S$-module of finite length, $\ell_S(M)=r<\infty$.
Under what ...
- 5,407
9
votes
3
answers
1k
views
Structure Theorem for finitely generated commutative cancellative monoids?
Is there a Structure Theorem for finitely generated commutative cancellative monoids?
Of course they can be densely embedded into a finitely generated abelian group, whose structure is known. Also, ...
CommunityBot
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1
vote
1
answer
274
views
Q-Divisor and Determinant Map on a Maximal Order
Given a smooth projective surface $X$, let $A$ be a sheaf of maximal orders in a division ring.
Let us for simplicity assume $A$ ramifies in one curve $C$ with ramification index $e$. Let $A^*$ be the ...
7
votes
1
answer
372
views
Simultaneously orthogonally transform two SPD matrices to tridiagonal form?
Supposing you have two SPD matrices $A,B\in\mathbb{R}^{n\times n}$ are there any known results on the existence or non-existence of a unitary matrix $Q$ such that $Q^\top A Q=T_A$ and $Q^\top B Q=T_B$ ...
0
votes
1
answer
352
views
Uniqueness of maximal subfields
Let D be a division ring with center Z. Let R and K be two maximal subfields of D, both purely inseparable of exponent one ( means the p power of each of them in Z).
Why are R and K isomorphic?
Or a ...
- 2,178
9
votes
1
answer
679
views
Mathematical software for computing in integral group rings of discrete groups?
I'm doing computations in the integral group ring of a discrete group,
in particular the discrete Heisenberg group. In this case elements
are integral combinations of monomials $x^k y^m z^n$, where ...
- 5,654
9
votes
1
answer
509
views
Maximal localizations of von Neumann algebras
Suppose M is a von Neumann algebra.
Denote by L its maximal noncommutative localization,
i.e., the Ore localization with respect to the set of all left and right regular elements,
i.e., elements whose ...
- 25.5k
3
votes
2
answers
1k
views
Spectral decomposition for an arbitrary linear combination of position and momentum operators
Suppose we have the Hilbert space L2(Rn) and we have n operators Qi and n operators Pi defined in the usual way by:
Qi ψ(q1,q2,...,qn) = qi ψ(q1,q2,...,qn)
Pi ψ(q1,q2,...,qn) = -i $\frac{...
- 6,342
2
votes
2
answers
302
views
How canonical is the triangular decomposition of a rational Cherednik algebra?
Introduction:
Let $V$ be a finite-dimensional $\mathbb{C}$-vector space, let $G \leq \mathrm{GL}(V)$ be a finite subgroup and let $\kappa:V \times V \rightarrow \mathbb{C}G$ be an alternating bilinear ...
- 44.7k
4
votes
0
answers
179
views
Global dimensions of orders over non-Gorenstein centers
This question concerns the following Lemma 4.2 in this paper by Van den Bergh:
Lemma: Let $R$ be local, normal Gorenstein ring of dimension $d$. Suppose $M$ is a reflexive $R$ module such that $A=\...
- 30.5k
21
votes
3
answers
2k
views
What is the precise relationship between groupoid language and noncommutative algebra language?
I have sitting in front of me two 2-categories. On the left, I have the 2-category GPOID, whose:
objects are groupoids;
1-morphisms are (left-principal?) bibundles;
2-morphisms are bibundle ...
CommunityBot
- 1
4
votes
0
answers
1k
views
Associative binary operations on natural numbers
Which are all the associative binary operations on natural numbers ?
Certain results in this regard can be found in arxiv:math/0508215.
It appears that such associative operations cannot grow too fast....
- 118k
1
vote
1
answer
262
views
$\omega$-monoids
Does the notion of $\omega$-monoid exist, analogous to the notions of $\omega$-groupoid and $\omega$-category? If so, some references would be appreciated.
This is an attempted rephrasing of question:...
CommunityBot
- 1
8
votes
1
answer
459
views
Semisimple-ish rings!
Let S be the class of all rings R which have 1 and satisfy this condition:
for every "non-zero" right ideal I of R there exists a "proper" right ideal J of R such that I + J = R. (The + here is not ...
- 6,734
10
votes
1
answer
746
views
Division algebras in which every proper subfield is maximal
I have a (noncommutative) division algebra D which is finite dimensional over its center F. I know that every subfield of D which contains F properly is a maximal subfield of D. What can we say about ...
- 2,178
3
votes
1
answer
350
views
Special subalgebras of central simple algebras
In this question F is a field and all algebras are finite dimensional F algebras.
Let X be the set of all F algebras A for which there exist an F algebra B and an F division algebra D such that F is ...
- 20.8k
17
votes
1
answer
3k
views
Do these conditions on a semigroup define a group?
As is well known, if $S$ is a semigroup in which the equations $a=bx$ and $a=yb$ have solutions for all $a$ and $b$, then $S$ is a group. This question arose when someone misunderstood the conditions ...
CommunityBot
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5
votes
0
answers
350
views
Chain/Hierarchy of Monoids
Let's assume that we have the following collection of structures:
Some space $P$.
Monoids $(M_{i+1},\circ_{i+1})$, and
Actions $\bullet_{i+1}:M_{i+1}\times M_i\to M_i$, for $i\ge 0$
And $\bullet_{0}:...
- 1,882
19
votes
4
answers
4k
views
What are your favorite finite non-commutative rings?
When you are checking a conjecture or working through a proof, it is nice to have a collection of examples on hand.
There are many convenient examples of commutative rings, both finite and infinite, ...
- 1,793