All Questions
393 questions with no upvoted or accepted answers
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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
votes
0
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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
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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
votes
0
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 ...
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2
votes
0
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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
2
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169
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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
2
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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 ...
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2
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91
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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
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0
answers
80
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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
2
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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. ...
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2
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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
2
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answers
62
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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) ...
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2
votes
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answers
73
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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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51
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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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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
2
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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
2
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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
2
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answers
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
2
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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
2
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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
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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
2
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180
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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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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
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answers
216
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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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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
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0
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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-...
2
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0
answers
122
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First-countable topological monoids without local absorbing elements whose topology is induced by a semimetric
This is a follow up of Question 163246. For the reader's convenience, let me first copy&paste some basic definitions.
We let a semimetric on a set $X$ be a function $d: X \times X \to [0,\infty]$ ...
- 10.5k
2
votes
0
answers
90
views
Poincaré Duality of a quasi-free algebra
I'm completely stumped on this one (yet I feel it is obviously true or obviously false)
If $A$ is a quasi-free algebra, then must it satisfy Poincaré duality?
All i need to find is a protective ...
2
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0
answers
66
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The norm of a polynomial f in a skew polynomial ring must be in the center
This is proved in Prop 1.7.1 in Jacobson's book ``Finite dimensional division algebras over fields". But I am not clear why the norm n(f), defined as the norm of the matrix representation of f by ...
- 43
2
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0
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124
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Reasoning about "approximately" associative structures and "almost monoids".
If $(M,+)$ is a monoid then it obeys the laws:
$$m_1 + 0 = 0 + m_1 = m_1$$
$$m_1+(m_2+m_3)=(m_1+m_2)+m_3$$
But what if I have a structure $(A,+)$ that is almost a monoid, but not quite. For example,...
- 21
2
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0
answers
272
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Continuity of multiplicative character
Let $G$ be a discrete group and $\beta (G)$ denote the Stone-Cech compactification of $G$, a right topological semigroup. By a multiplicative character, I mean a mapping that preserves multiplication ...
- 61
2
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0
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91
views
Fat modules on some algebras.
Let $A$ be a graded $k$-algebra and $M$ a graded right $A$-module. $M$ is called a fat $A$-module if it is generated by degree $0$ and has constant Hilbert polynomial $2$. I wonder for which finitely ...
- 21
2
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0
answers
63
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Boundedness of modules on AS regular algebras
Let $k$ be an algebraically closed field and $A$ be an Artin-Shelter regular $k$-algebra. Fix a numerical polynomial $H(t)$. I would like to know whether or not semi-stable f.g. graded $A$-modules ...
- 1,663
2
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0
answers
137
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Noncommutative Castelnuovo-Mumford regularity
I am looking for noncommutative version of Castelnuovo-Mumford regularity. To be more precise, let $A=\oplus_{i=0}^{\infty}A_{i}$ be a $good$ (finite global dimension, connected etc) noncommutative ...
- 21
2
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196
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Recursion formula for odd holonomic function
Let $E$ be the space of maps
$f:\mathbb{Z}\longrightarrow\mathbb{C}[t^{\pm}]$
satisfying $\forall i$, $f(i)=-f(-i)$.
The algebra
$$\mathcal{T}=\mathbb{C}[t^{\pm}]\langle L^{\pm},M^{\pm}\rangle/(LM-...
- 243
1
vote
0
answers
128
views
What is a quantum condensed space?
Due to a categorical equivalence involving compact topological spaces and unital commutative $\mathrm{C}^*$-algebras, there is a practise involved in so-called quantum mathematics where a ...
- 1,037
1
vote
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answers
30
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Star-algebra isomorphism
I have asked this question: When an algebra isomorphism preserves positive involution, but now I want to modify it.
Let $A$ and $B$ be $K$-algebras where $K$ is a field with a unique ordering.
We say ...
- 143
1
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0
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90
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Multiplicative bases, path algebras, and Ext algebras
I am interested in understanding when a multiplicative basis exists for finite dimensional algebras over an algebraically closed field, and, in particular, Ext-algebras that are finite dimensional.
It ...
- 71
1
vote
0
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69
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On a lemma of projective dimension
Let $R$ be a finite-dimensional algebra, and $A=R\oplus A_1\oplus A_2\oplus \dotsb$ be an $\mathbb{N}$-graded algebra which is locally finite (i.e. all $A_i$'s are of finite dimension). Let $\text{...
- 157
1
vote
1
answer
84
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Simple-direct-injective modules
A right $R$-module $M$ is called a simple-direct-injective module if it satisfies any of the following equivalent conditions:
For any simple submodules $A,B$ of $M$ with $A \cong B \subseteq^{\oplus} ...
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1
vote
0
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86
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Inner product on Standard form of von Neumann algebra
Let $(M, H, H_+,J)$ be a standard form of a von Neumann algebra $M$ acting on a complex Hilbert space $H$ endowed with a self-dual cone $H_+$. Is it true that
$$\langle x,yz\rangle=\langle zx,y\rangle$...
- 131
1
vote
0
answers
35
views
Number of right divisors of a central skew polynomial
Let $\mathbb{F}$ be a finite field of $p$ elements, $\sigma \in \operatorname{Aut}(F)$ of order $m$, $\mathbb{F}^\sigma$ be the fixed field of $\sigma$, and $\mathbb{F}[x,\sigma]$ be a skew polynomial ...
- 323
1
vote
0
answers
69
views
Structure of tame concealed algebra of Euclidean type
I wanted to know some references where people have studied the representation theory of tame concealed algebra of Euclidean type. What do we know about the structure of their module category? What ...
- 839
1
vote
0
answers
108
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Does the center of any finitely generated associative algebra over a field have finite type?
Consider the monoid algebra $R:=K\langle x_1,\dots,x_n\rangle$ generated by $n$ letters $x_1,\dots,x_n$ for $n>1$ over field $K$. Equivalently, $R$ is the tensor algebra $T(V)$ on the $n$-...
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1
vote
0
answers
48
views
Neighborhoods of idempotents in topological inverse semigroups
In a topological group, for any neighborhood $U$ of the origin, there is another such neighborhood with the property that $V.V\subseteq U.$ I conjecture a similar property for topological inverse ...
- 1,093
1
vote
0
answers
95
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Eventual stabilization for repeatedly adding multiplayer games
This question is an outgrowth of a couple previous questions of mine. In order: 1,2,3. This should be fully self-contained, but those questions may help motivate this one.
To keep things readable, I'...
- 20.5k
1
vote
0
answers
91
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A term for a submonoid of a free abelian monoid?
Are there multiple ways of characterising which monoids are submonoids of free abelian monoids?
What free abelian monoids are:
A free abelian monoid $\mathbb N^d$ with $d$ generators (where $d$ is an ...
- 4,943