All Questions
1,123 questions
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230
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Reference request: the UEA of the LR-algebra of tangent vector fields on a smooth manifold coincides with the derivation ring and the ring of diff ops
Let $\mathcal{M}$ be a smooth real manifold and let $A:= \mathcal{C}\left(\mathcal{M}\right)$ be the real algebra of smooth functions on $\mathcal{M}$.
Recall from McConnell, Robson, Noncommutative ...
- 718
2
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90
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On the use of the fundamental exact sequence of K\"ahler differentials in a paper of Lyubeznik
Let $k$ be a field, $R := k[x_1, \cdots , x_n]$ the polynomial ring in $n$ indeterminates over $k$ and $f$ a nonzero element of $R$. The following paper of Lyubeznik which I have been recently reading,...
- 81
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answers
74
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Terminology and notation for generated subgroups
I would like to think about formation of the smallest subgroup (or monoid, or whatever) $H$ of $G$ containing two given subgroups $A$ and $B$ as an operation on subgroups, and I wonder if there is a ...
- 12.5k
2
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answers
270
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Road map: beyond Artin-Wedderburn theorem
For a noncommutative semisimple ring $R$, its structure and its category of representations can be largely understood using Artin-Wedderburn theorem. Such structure theory is useful, for example, in ...
- 5,230
2
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89
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Semigroups associated to binary necklaces and their semigroup algebra
I came across the following semi-group and the associated finite dimensional semi-group algebras over a field $K$ (which are Nakayama algebras) as they have very nice homological properties. My ...
- 26.5k
2
votes
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96
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Non-commutative version of the order dimension of a poset
I view the order dimension of a poset $P$ as an inherently commutative notion. On the one hand, it can be defined via realizers, which I find fairly intuitive from an order-theoretic viewpoint. On the ...
- 51
2
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answers
50
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Existence of nontrivial transfinite divisibility in $R$-modules
Let $R$ be a unital, possibly noncommutative ring and $s \in R$. For a right $R$-module $M$, define $Ms = \{ms \mid m \in M\}$; this is an additive subgroup of $M$, which is a module over the ...
- 63.9k
2
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132
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Rings whose finitely-generated modules are co-hopfian
Let $A$ be a unital, possibly noncommutative ring. Dischinger showed [1] that the following are equivalent:
For every $a \in A$, there exists $n \in \mathbb N$ such that $a^n A = a^{n+1} A$;
For ...
- 63.9k
2
votes
0
answers
98
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Dimension of center of $k[G]/\mathrm{rad}k[G]$ when characteristic of $k$ divides the order of $G$
Let $G$ be a finite group and consider $k[G]$ where $k$ is a field. In the scenario where $\mathrm{char}(k)$ divides $|G|$, how can one show that the dimension of $Z(k[G]/\operatorname{rad}k[G])$ is ...
2
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116
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Bijection between monoid of nilpotent decreasing self-maps and local subsemigroup
Let $\mathcal{C}_X\cong\mathcal{C}_n$ be the monoid of self-maps $\alpha$ of $X=\{1\dots,n\}$ that are order-preserving ($\forall x,y$, $x\le y$ $\Rightarrow$ $\alpha(x)\le\alpha(y)$) and decreasing ($...
- 355
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answers
33
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On the number of connected functional digraphs recoverable from the preimage set size structure
I am studying the list of inverse images (preimage sets) of some function $f$ at a given inverse depth $j$ -- for each element $x_i$ of a finite domain $X$.
For example,
$P_j=\left[f^{-j}(...
- 23
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0
answers
91
views
Is the natural action of the monoid of endomorphisms is a complete invariant for group?
Let $\alpha$ and $\beta$ be actions of semigroups $A$ and $B$ on sets $X$ and $Y$ respectively. Recall that $\alpha$ and $\beta$ are called isomorphic if there exists an isomorphism $\phi$ between ...
- 6,962
2
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56
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Non-singular rings which are Rickart
A ring $R$ is said to be a right Rickart ring if the right annihilator of any element in $R$ is of the form $eR$ for some idempotent $e \in R$.
It turns out that a ring $R$ is right Rickart iff every ...
- 1,065
2
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48
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Compute irreducibles of monoid
Given $n > 0$ and $w \in \mathbb{Z}^n$. Is there an efficient algorithm to compute the set of irreducible elements of the monoid $M_w = \{x \in \mathbb{N}^n \mid \langle x,w\rangle = 0 \}$?
Here, ...
2
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70
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Embedding problems on quantum groups？
We work over the field of complex numbers.
We have known that Lie algebra of type $A_2 $is a subalgebra of type $G_2$. However, when we consider their quantum groups, is this true i.e. does there ...
- 165
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60
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Integrals in noncommutative graded algebras which are not necessarily Hopf
Let $\mathbf{k}$ be a field. Let $A$ be a finite dimensional $\mathbb{Z}_{\geq 0}$-graded $\mathbf{k}$-algebra such that $A^0=\mathbf{k}1$. Let $m$ be the maximal non-negative integer such that $A^m\...
- 1,066
2
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answers
60
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Are there finitely-presented astral monoids?
We say a semigroup $S$ is $k$-astral if there exists a finite set $F \subset S$ such that
whenever $s_1, s_2, ..., s_k \in S$ there exists $s \in S$ such that $\forall i: s_i \in sF$. Say $S$ is ...
- 6,652
2
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answers
103
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Lattices with trivial coinvariants for finite groups
Let $G$ be a finite group. A $\mathbb{Z}G$-lattice is a $\mathbb{Z}G$-module that is (as abelian group) a free abelian group of finite rank.
Question: Is there a finite group $G$ and a $\mathbb{Z}...
- 2,160
2
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42
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Concerning $(x,y) \mapsto (x^{\frac{n}{r}+1}y + A,\mu x^{-\frac{n}{r}}+B)$
Let $r \in \mathbb{N}-\{0\}$.
Commutative case:
Let $f : (x,y) \mapsto (p,q)$ be a map from $\mathbb{C}[x,y]$ to $\mathbb{C}[x^{1/r},x^{-1/r},y]$ satisfying the following two conditions:
(i) $\...
- 2,837
2
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answers
44
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Partially commutative elements in powers of augmentation ideal
Let $\vartheta$ a relation of parcial commutation over a set $X,$ and consider the respective free parcially commutative group $F(X, \vartheta).$ Let $K[F(X, \vartheta)]$ the parcially commutative ...
- 121
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answers
92
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Monoidal structure on left dg-modules over a brace algebra
Relating to my other question: Modules over Hopf Algebras and $E_2$-algebras
Preliminary: Let $A$ be an associative dg-algebra that is also an algebra over the brace operad. Let $M$ and $N$ be left ...
- 527
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answers
68
views
Transmission of finite projections
Let $A$ be a Baer*-ring. Let us denote $L(x)$ by the left projection of $x$ (the smallest projection with $L(x)x=x$).
Let $p$ be a finite projection in $A$. Is $L(xp)$ a finite projection for every $...
- 4,058
2
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0
answers
153
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Algebraic version of unilateral shift
It was confirmed that Wold-type decomposition can be extended from von Neumann algebras to Baer*-rings (see this paper). By algebraic tools the notion of unilateral shifts is successfully transmitted ...
- 4,058
2
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0
answers
80
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Are the roots of an infinitely divisible probability infinitely divisible themselves?
Let $\mu$ be an infinitely divisible probability on a topological group $G$. If $\nu ^{* n} = \mu$ for some $n$, is $\nu$ an infinitely divisible probability too?
A sufficient criterion would be to ...
- 5,407
2
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answers
81
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A semigroup property related to von Neumann regularity
A very common and useful notion in rings is that of von Neumann regular elements: those elements $a\in R$ such that there exists $b\in R$ satisfying $aba=a$. As this is a property defined solely ...
- 18.7k
2
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50
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Nonautonomous wave equation of memory type
I want to apply the semigroup approach of nonautonomous evolution equation for the following wave equation
$$u'' - \Delta u + \int\limits_0^t {g(s)} \Delta u(s)ds = 0$$
This problem can be written ...
- 617
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answers
169
views
What is the difference between a monosemiring and a semigroup?
What is the difference between a monosemiring and a semigroup?
The following definitions are for clarity of my question.
A semigroup $S$ is a non empty set that satisfies closure and associativity ...
- 203
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answers
63
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QF-3 monoid algebras
A finite dimensional algebra $A$ is called QF-3 in case the injective envelope of the regular module is projective. For example all Frobenius algebras are QF-3.
Given a monoid algebra $kG$ of a finite ...
- 26.5k
2
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0
answers
91
views
Semigroups of nondecreasing functions
Consider some partially ordered set $(E,\leq)$. Assume either that it is countable with the discrete topology, or that it has some topology compatible with the order, preferably one that makes it into ...
2
votes
0
answers
80
views
group action on Tor groups of modules and smash product
I am trying to understand theorem 3.4.2 from the paper "Bernstein-Gelfand-Gelfand complexes and cohomology of nilpotent groups over $\mathbf Z_{(p)}$ for representations with $p$-small weights" by ...
- 391
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98
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If $H$ is an atomic, unit-cancellative monoid such that the set of atoms of $H$ is finite up to associates, then $H$ is BF
In a previous version of this post, $H$ was an atomic commutative monoid such that the quotient $H/H^\times$ is finitely generated, and I was asking if such conditions were enough for $H$ to be BF. ...
- 10.5k
2
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0
answers
206
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Is there an anti-commutator analog of Zassenhaus formula?
Is anyone familiar with an anti-commutator analog Zassenhaus formula? I have been able to find the anti-commutator analog of the BCH formula
$$e^ABe^A= B + \{B,A\}+\frac{1}{2!}\{\{B,A\},A\}+ \frac{1}{...
- 21
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answers
62
views
Extensions of an ideal-theoretic criterion for a monoid to be BF
Let $H$ be a multiplicatively written, commutative monoid. We denote by $H^\times$ the set of units (or invertible elements) of $H$, and by $\mathcal A(H)$ the set of atoms (or irreducible elements) ...
- 10.5k
2
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73
views
injective dimension of envelope algebras
Let $A$ be a connected graded algebra and $A^o$ the opposite algebra of $A$. Let $A^e=A\otimes A^o$. Suppose that $A$ has a finitely generated projective resolution as a graded $A^e$-module.
My first ...
- 564
2
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answers
51
views
Existence of a canonical embedding from a quotient of $\mathscr{F}^\ast(\mathcal A(H))$ to another
Let $H$ be a multiplicative, commutative monoid, and denote by $H^\times$ the group of units of $H$, by $\mathcal A(H)$ the set of atoms (or irreducible elements) of $H$, by $\mathscr{F}^\ast(\mathcal ...
- 10.5k
2
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0
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139
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A certain non-clean ring
I am searching for a non-commutative ring $R$ with identity such that $R$ is not a clean ring and $R/Soc(R_R)$ is a Boolean ring. By a clean ring I mean a ring each of whose elements is a sum of a ...
- 355
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135
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Is a ring with stable range 2 2-Hermite?
Let $R$ be a (possibly non-commutative) ring. The left stable range of $R$ (denoted $sr_l(R)$) is the smallest $n$ such that every left unimodular row of length $>n$ is reducible. A similar ...
- 61
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106
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Descent of flatness from algebras to monoids II
This is a follow-up question to this one. There, Benjamin Steinberg showed that a morphism of commutative monoids $u$ need not be flat if the induced morphism of $R$-algebras $R[u]$ is flat for some ...
- 6,700
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227
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What is the motivation behind the definition for a smooth differential graded category?
Let $\mathcal{A}$ be an $\mathbb{F}$-linear differential graded category. It is said to be smooth if it is a perfect complex over the differential graded category $\mathcal{A}^\circ\otimes_\mathbb{F}\...
- 1,716
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448
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Completion of an algebra
Based on arXiv:math/9802041v1, there is a definition for $NC$-filtration and $NC$-completion of an associated algebra over the complex numbers:
Let $R$ be an associative algebra and $R^{\rm Lie} = (R,...
- 51
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0
answers
178
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Monoid prime ideals and prime congruences
I was wondering what the connection is between the notion of "prime congruence" on a monoid, and the notion of "prime ideal" in a monoid. Starting from a prime ideal $P$ in a monoid $M$, one can ...
- 4,547
2
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193
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Factorisation of twisted polynomials
Let $K=\mathbb{C}((t))$ and let $K_m=\mathbb{C}((t^{1/m}))$. let $K\{x\}$ denote the ring of twisted polynomials. The addition in this ring is defined as usual, but the multiplication is adjusted by ...
- 2,751
2
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87
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Terminology for torsion semigroups where the order of elements is uniformly finite
A (multiplicatively written) semigroup $\mathbb A = (A, \cdot)$ with the property that ${\rm ord}_\mathbb{A}(a) := |\{a^n: n \in \mathbf N^+\}| < \infty$ for every $a \in A$ is called a periodic (...
- 10.5k
2
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203
views
Profinite Topology
Let $V$ and $W$ be pseudovarieties of finite groups. For a finite inverse monoid $M$, the $V$-kernel of $M$ is defined to be the intersection of all sets $f^{−1}(1)$, $f$ is a relational morphism ...
- 421
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180
views
Pro-p topology on free group
Let $H$ be a finitely generated subgroup of the free group $F(A)$ and $G_P$ the pseudovariety of all finite $p$-group with $p$ fixed prime number. We endow $F(A)$ with the pro-$G_p$ topology. Suppose ...
- 421
2
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139
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Goldie's Theorem for Semigroups
Goldie's theorem is a theorem in noncommutative ring theory that gives a clear picture of semiprime Noetherian rings (actually a slightly broader class). Let $R$ be a semiprime Noetherian ring. The ...
- 6,870
2
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0
answers
99
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Turning left modules into right modules over a homotopy Gerstenhaber algebra
For simplicity's sake, let $A$ be a dg-algebra over $\mathbb{Z}/2\mathbb{Z}$.
In the case when $A$ is a commutative algebra, we can turn a left $A$ module into a right $A$ module trivially. Of course ...
2
votes
0
answers
216
views
Standard name for a Monoid/Semigroup with $a+b \leq a, b$?
I have seen suplattice and inflattice being used when dealing with a lattice. What about when you don't have a lattice?
For instance, for reals $a,b > 0$, define $$a \oplus b = \frac{1}{\frac{1}{a}...
- 121
2
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0
answers
1k
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Tensor product of commutators vs. commutator in a tensor product
Let $R$ be a (noetherian) commutative ring, and let $V$ and $W$ be finitely generated free $R$-modules. Let $X \subseteq \mathrm{End}_R(V)$ and $Y \subseteq \mathrm{End}_R(W)$ be finite subsets, and ...
- 4,015
2
votes
0
answers
91
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Algorithms to find the solutions of a homogenous matrix equations for non-commutative rings
In one paper from 1980 I found a note that there are no known algorithms for solving
homogenous matrix equations $x \cdot M = 0$ for matrices which elements belong to a non-commutative ring.
(The non-...
