All Questions
1,123 questions
6
votes
1
answer
214
views
Weak associativity
Let $(V,*)$ be an algebra and denote $A_*\in \text{Hom}(V^{\otimes 3},V)$ the associator of the binary product $*\in \text{Hom}(V^{\otimes 2},V)$ defined as $A_*(a,b,c):=(a*b)*c-a*(b*c)$.
The ...
- 16.9k
2
votes
0
answers
174
views
Moduli spaces of stable sheaves on noncommutative projective schemes
In noncommutative algebraic geometry in the sense of Artin and Zhang, can we construct moduli spaces of stable sheaves on noncommutative projective schemes as (commutative)schemes ?
I would appreciate ...
- 889
6
votes
3
answers
411
views
Problem 0.9.10 in Cohn's "Free Ideal Rings and Localization in General Rings" (CUP, 2006)
Let $S$ be a monoid. On p. xvii of P.M. Cohn's Free Ideal Rings and Localization in General Rings (CUP, 2006), one reads that
an element $u \in S$ is regular if (quote) "[...] it can be ...
- 10.5k
24
votes
5
answers
2k
views
Lie groups vs Lie monoids
Does there exist a well developed theory of a class of objects which might rightfully be called Lie monoids? By this I mean with axioms similar to those of Lie groups, but with the axiomatic existence ...
- 2,369
9
votes
7
answers
2k
views
Hochschild/cyclic homology of von Neumann algebras: useless?
Hochschild homology gives invariants of (unital) $k$-algebras for $k$ a unital, commutative ring. If we let our algebra $A$ be the group ring $k[G]$ for $G$ a finite group, we get group homology. ...
- 63.9k
5
votes
1
answer
175
views
Finding non-inner derivations of simple $\mathbb Q$-algebras
What's a good example of a simple algebra over a field of characteristic $0$ which has a non-inner derivation but also has the invariant basis number property (IBN)?
I'm under the impression that when ...
- 38.6k
2
votes
0
answers
108
views
Left-elements of a numerical semigroup generated by two elements
A numerical semigroup $S$ is a semigroup in $\mathbb{N}$ such that $\mathbb{N}\backslash S$ is finite. It is known that there exists always a set $M$ such that an element in $S$ can be expressed as a ...
- 115
9
votes
1
answer
743
views
Why is choice needed in Ellis' Lemma?
Ellis Lemma on idempotent elements asserts that:
Lemma (Ellis). Every compact semigroup has an idempotent.
The proof below is excerpted from Todorcevic's Introduction to Ramsey Spaces, Lemma 2.1.
...
- 16.4k
1
vote
1
answer
90
views
Affine semigroup generating a lattice
This is a cross-post from MSE.
Everything is assumed to be finite-dimensional. Let $S$ be a finitely generated affine semigroup (i.e. a subsemigroup of a lattice $N$ of a Euclidean space). Assume that ...
- 63.9k
7
votes
1
answer
350
views
Pushouts of injective monoid homomorphisms
Given a pushout square in the category of monoids
$$\begin{array}{ccc}A & \rightarrow & M \\ \downarrow && \downarrow \\ N & \rightarrow & P\end{array}$$such that $A \to M$ and ...
- 63.9k
1
vote
0
answers
156
views
Pseudo-coherent complexes over sheaves of non-commutative rings
I am posing a question on derived categories to which I was not able to find an answer anywhere in the literature. I would appreciate any answer, hint or suggestion.
Assume that $\mathcal{R}_X$ is a ...
- 609
2
votes
1
answer
67
views
$E$-separated semigroups
Definition. A semigroup $X$ is called $E$-separated if for any distinct idempotents $x,y\in X$ there exists a homomorphism $h:X\to Y$ to a semilattice $Y$ such that $h(x)\ne h(y)$.
Observe that $X$ is ...
- 41.9k
3
votes
0
answers
88
views
Explicit separability idempotent for the center of a separable algebra
Let $A$ be a $k$-algebra for some commutative ring $k$. Recall that $A$ is said to be separable over $k$ if the multiplication map $A\otimes_k A^{\operatorname{op}}\to A$ has a section as a map of $A\...
- 15.8k
2
votes
1
answer
167
views
Existence of reduced norms for CSAs using fpqc descent
Let $k$ be a field and $A$ be a central simple algebra over $k$. It's known that $A$ has a splitting field (i.e. a field $K/k$ such that $A_K\cong M_n(K)$ for some $n$) which is finite and Galois.
...
- 12.9k
0
votes
2
answers
219
views
Are there any central simple algebras admitting a standard basis?
Are there any central simple algebras admitting a standard basis?
By a standard basis I mean a normal basis that has a cyclic property generalizing that of the familiar basis $1, i, j, k$ for ...
- 12.9k
4
votes
0
answers
166
views
A non-commutative, left duo ring whose only unit is the identity
Let $R$ be a ring (here, rings are always associative, unital, and non-zero). We say that $R$ is a left duo ring if $aR \subseteq Ra$ for every $a \in R$.
Question. Is there a non-commutative, left ...
- 10.5k
30
votes
4
answers
3k
views
A mysterious Heisenberg algebra identity from Sylvester, 1867
I am trying to understand two papers by James Joseph Sylvester:
P92: "Note on the properties of the test operators which occur in the calculus of invariants, their derivatives, analogues, and laws of ...
- 10.5k
5
votes
1
answer
266
views
Rings s.t. each element has a power lying in the center (and their completely prime ideals)
Let $R$ be a ring (throughout, all rings are associative and unital). We say $R$ satisfies condition (C) if, for every $a \in R$, there exists an integer $n \ge 1$ (depending on $a$) such that $a^n$ ...
- 12.9k
5
votes
0
answers
241
views
Non-commutative rings where every non-unit is contained in a completely prime ideal
Below, all rings are associative and unital; and the word "ideal" always refers to a two-sided ideal.
Let's stipulate that a ring $R$ has property (P) if every non-unit of $R$ is contained ...
- 10.5k
11
votes
0
answers
214
views
Is it decidable if a tree-presented semigroup contains an idempotent?
A semigroup presentation $\langle A | R\rangle$ is called tree-like if every relation has the form $ab=c$, $a,b,c$ are in $A$ and if two relations $ab=c, a'b'=c'$ belong to $R$, then $c=c'$ if and ...
- 4,713
7
votes
1
answer
281
views
Question concerning the coefficients of block idempotents
Let $G$ be a finite group. Let $p$ be a prime number such that $p \mid |G|$.
Let Irr$(G)$ denote the set of ordinary irreducible characters of $G$.
For $\chi\in$ Irr$(G)$ define $e_{\chi} := \frac{\...
- 44.4k
8
votes
0
answers
285
views
Matrix decompositions as monoid isomorphisms. Ever considered before?
I've noticed some correspondences between some matrix decompositions and monoid isomorphisms (always to some free commutative monoid), in addition to the one I asked about in a previous question:
...
- 12.9k
7
votes
0
answers
221
views
Strange formula for the dimension of a certain space of noncommutative polynomials
Consider a vector space $V_r(n)$ spanned by (noncommutative) monomials in variables $x_1,\ldots,x_r$
$$
x_{1}^{n_1}x_{2}^{n_2}\ldots x_{r}^{n_r}
$$
of total degree $n.$ Inside this space consider a ...
- 4,316
1
vote
0
answers
43
views
Interleaving in Viennot's Heaps models?
I am looking for past results on interleaving of heaps (in the sense of Viennot). For a very simplified example, suppose I have two pieces, (b1 a1 b1), and (b2 c2 b2), where the letter represents a ...
- 11
13
votes
1
answer
1k
views
For what sets $X$ do there exist a pair of functions from $X$ to $X$ with the identity being the only function that commutes with both?
It is not too difficult to show that if $X$ is an infinite set, then there exists a two-element subset of the group $\operatorname{Sym}(X)$ with trivial centralizer iff $\lvert X\rvert \leq \lvert\...
- 32.5k
9
votes
4
answers
954
views
Morita equivalence and moduli problems
Two rings $A$ and $B$ are said to be Morita equivalent if the category of modules over $A$ and $B$ are equivalent as additive categories. (Here I'm considering left modules).
Ex: $M_n(R)$ (the algebra ...
- 35.5k
4
votes
1
answer
368
views
Possible values of symmetric functions evaluated on quaternions
$\DeclareMathOperator\sym{sym}$Let $i$, $j$, $k$ be the units of quaternions, in particular $i^2=j^2=k^2=-1$, $ijk=-1$.
We will use non commutative variables $x$, $y$, $z$. Define $\sym_{a,b,c}$ to be ...
CommunityBot
- 1
2
votes
0
answers
73
views
Nonzero idempotents in compact semitopological semigroups with zero
Let $S$ be a compact semitopological semigroup. Then, $S$ contains minimal idempotents by Ellis' theorem.
Ellis' Theorem: In a compact left-topological (resp. right-topological) semigroup, every ...
- 2,605
2
votes
0
answers
145
views
Semigroup ideals of a ring or an algebra
Let $R$ be a ring or an algebra. Suppose $B\subseteq R$ satisfies the property $BR \subseteq B$ and $RB\subseteq B$. Is there a general theory of subsets with this property in a ring (resp. algebra) ...
- 63.9k
2
votes
0
answers
119
views
The "matrix direct sum" monoid modulo unitary equivalence
Given a commutative $*$-ring $(R,*)$, let $M(R,*)$ be the monoid whose elements are matrices over $R$ of all possible shapes and entries, including those that have $0$ columns or $0$ rows. Let the ...
- 4,943
7
votes
0
answers
194
views
Factoring a function from a finite set to itself
Let $S$ be a finite set and $f: S \to S$ be a function. Let $k = |f(S)|$ and let $\alpha$ be the partition of $S$ into $f$-fibers, i.e. $\alpha = \{ \alpha_t \}_{t \in f(S)}$ where $\alpha_t = f^{-1}(\...
- 695
4
votes
1
answer
446
views
What is a "cusp" ("кусок") in relation to Guba's embedding theorem?
I'm confused by the definition of a "cusp" as found in
V.S. Guba, Conditions for the embeddability of semigroups into groups, Math. Notes 56 (1994), Nos. 1-2, 763-769 (link).
In the words of Mark ...
- 10.5k
9
votes
2
answers
2k
views
A ring for which the category of left and right modules are distinct
What is an example of a ring $R$ for which the abelian category of left $R$-modules is not isomorphic to the category of right $R$-modules?
- 38.6k
3
votes
1
answer
157
views
Is it true that the structure of a commutative ordered semiring is unique on a commutative ordered monoid?
Is it true that the structure of a commutative ordered semiring with identity is unique on a commutative ordered monoid (i.e., the structure of the monoid and the order are consistent)? I am not ...
- 18.7k
1
vote
0
answers
85
views
Exponential of a sum in a non-commutative graded algebra
Let $a,b$ be two elements of a graded algebra $A$ such that $\deg(a)=1$, $\deg(b)=0$ and $[a,b]\neq 0$.
I would like to know whether there exits an explicit expression for the degree 1 component
$$\...
- 789
3
votes
1
answer
299
views
RIng that is flat over a subring as a right module but not as a left module
What is an example of a ring $R$ and a subring $S \subseteq R$ such that $R$ is flat as a right module but not flat as a left module.
The following question is my motivation:
Faithful flatness for ...
- 10.8k
4
votes
0
answers
363
views
A projective module over a domain that is not faithfully flat?
Let $R$ be a (noncommutative) unital ring which is a domain and let $\mathcal{N}$ be a non-zero projective (right) module. Projectivity of course implies that $\mathcal{N}$ is flat, but does the fact ...
- 1,766
4
votes
1
answer
417
views
Is a solvable group satisfying a semigroup law?
Let $S$ be the free semigroup on the set $\{x_1,\ldots ,x_n\}$, where $n$ is a positive integer. Suppose that $\mu=\mu (x_1,\ldots ,x_n)$ and $\nu = \nu (x_1,\ldots ,x_n)$ are two elements in $S$. We ...
- 883
8
votes
1
answer
645
views
Isomorphic morphisms. A 27-morphism category
Two morphisms of category $\ \mathbf C\ $ are isomorphic to one
another $\ \Leftarrow:\Rightarrow\ $ they are the opposite edges that are drawn horizontally (aimed East) of a commutative square that ...
- 4,786
0
votes
1
answer
139
views
Computationally intractable orbit of a monoid action on a finite set
Suppose for each integer $n\geq 1$ we have a submonoid $M_n\subset \mathrm{Self}(\{1, \dots, n\})$ of self-maps of $\{1, \dots, n\}$.
A characterization of $M_n$ is an algorithm that takes an integer $...
- 7,952
2
votes
2
answers
181
views
Module complements to rings embedded in a projective module
Let $R$ be noncommutative unital ring and $M$ a projective (right) $M$-module. Assume that $R$ embedds into $M$ as a right -module.
A) If $R$ is a semisimple ring, then of course $R$ admits an $R$-...
- 38.6k
2
votes
0
answers
203
views
Can the relation between count of commuting pairs and conjugacy classes for finite groups be generalized to semigroups?
It is well-known that number of pairs of commuting elements in finite group G is equal to number of conjugacy classes multiplied by cardinality of G.
More generally here (MO275769) Qiaochu Yuan ...
- 24.9k
2
votes
1
answer
166
views
Submonoid of free monoid with certain properties
Let $N$ be a submonoid of a free monoid $M$ such that
$m_1nm_2\in N \Rightarrow m_1,m_2\in N$ for any $m_1,m_2\in M$ and $n\in N\setminus\{1\}$. $\quad\quad\quad\quad$ (C)
Do such submonoids ...
- 491
8
votes
1
answer
452
views
Separable and finitely generated projective but not Frobenius?
Let R be a commutative ring, and $A$ an $R$-algebra (possibly non-commutative). Then $A$ is separable if it is finitely generated (f.g.) projective as an $(A \otimes_R A^{\mathrm{op}})$-algebra. ...
- 63.9k
9
votes
0
answers
347
views
What is the precise connection between logarithmic algebraic geometry and the field with one element?
Monoid schemes (a.k.a. $\frak M$-schemes) have been introduced by Deitmar as a possible approach to geometry over the field with one element. These build upon monoids as the basic building blocks for ...
- 11.8k
1
vote
0
answers
88
views
Sequences generated from commuted quaternions and general commuted linear transformations
Given a pair of non-commuting linear transformations, $A$ and $B$, define the "next
pair" in a sequence as $A*B$ and $B*A$. I am interested in finite cycles (i.e.,
the sequence eventually ...
- 1,547
8
votes
1
answer
238
views
Functions over monoids which factor in two different ways
This is a follow-up question to this MO question, which was asked by Richard Stanley in a comment to my answer there.
Let $S$ be a commutative monoid and $f(x_1, \dots, x_n)$ be a function from $S^n$ ...
- 1,916
1
vote
3
answers
585
views
Terminology for certain monoids which are to monoids like fields are to rings
Let $M$ be a commutative monoid with zero. Then the condition $M^* = M \setminus \{0\}$ is very similar to the condition for a commutative ring to be a field. This analogy is also used in the work "...
- 11.8k
26
votes
3
answers
724
views
Subtraction-free identities that hold for rings but not for semirings?
Here is a concrete, if seemingly unmotivated, aspect of the question I am interested in:
Question 1. Let $a$ and $b$ be two elements of a (noncommutative) semiring $R$ such that $1+a^3$ and $1+b^3$ ...
- 12.9k
2
votes
1
answer
287
views
Every module of finite uniform dimension is a direct sum of (finitely many) indecomposable
Crossposted on StackExchange on July 28 (no answer so far).
Let $R$ be a (commutative or non-commutative, associative, unital) ring. It is well known that any artinian or noetherian $R$-module $M$ can ...
- 18.7k