All Questions
1,123 questions
5
votes
0
answers
86
views
von Neumann regular ring homomorphisms
Let us call a ring homomorphism $f\colon R\rightarrow S$ von Neuman regular if it has the property that for every left $S$-module $M$, the left $R$-module $f^*M$ is flat.
In particular, $\mathrm{id}...
- 858
5
votes
0
answers
219
views
Constructing a noncommutative algebra from a commutative algebra
I was told at a conference that one way to construct a noncommutative algebra from a commutative one is to "replace the product of finite spaces (which on the level of continuous functions corresponds ...
- 111
5
votes
0
answers
104
views
Finitely generated submodules of projectives lie inside f. g. projectives?
Let $R$ be a (not necessarily commutative) ring.
If $M$ is a finitely generated submodule of a projective module $P$, is there a finitely generated projective submodule $P'$ such that
$M \subseteq P'...
- 51
5
votes
0
answers
208
views
Which groups can occur as the group of units of finite-dimensional noncommutative algebras?
This is a continuation of a previous question: Connectedness of groups of units in finite-dimensional commutative algebras.
Let $k$ be an algebraically closed field of characteristic $0$. Which ...
- 7,127
5
votes
0
answers
64
views
Characters on monotone functions
Characters on the semigroup $(C_{+}^{b}(\mathbb{R}^{d}),+)$, i.e. on bounded positive continuous functions with the ususal pointwise addition, are known to be of the form $C_{+}^{b}(\mathbb{R}^{d})\ni ...
- 289
5
votes
0
answers
395
views
Derived tensor products and Tor of commutative monoids
Two commutative monoids $M,N$ have a tensor product $M\otimes N$ satisfying the universal property that there is a tensor-Hom adjunction for any other commutative monoid $L$: $$\text{Hom}(M\otimes N,L)...
- 889
5
votes
0
answers
264
views
Do almost commutative flat degenerations induce equality in K-theory? (Or: Is the characteristic variety actually a support of a class in $K$-theory?)
I intentionally phrased the title to match a different question which is almost identical to the one i'm asking. However similar, the answer there, which uses commutative algebraic geometry, is not ...
- 7,789
5
votes
0
answers
99
views
Zappa-Szép products of the monoid of integers with itself
Question
What are all the functions $\alpha , \beta : \mathbb{N} \to \mathbb{N}$ satisfying the following functional equations?
$\bullet ~~~~ \alpha(0)=0, \quad \beta(0)=0\\
\bullet ~~~ \...
- 5,482
5
votes
0
answers
637
views
Unique product groups (and semigroups)
A group $G$ is called a u.p.-group (short for unique product group) if for all nonempty finite subsets $A,B\subseteq G$, there exists an element $g\in A \cdot B$ which can be uniquely written as a ...
- 18.7k
5
votes
0
answers
317
views
Hochschild Cohomology of the Quantum Torus
I would like some advice on how to compute directly, or by a higher powered method the Hochschild Cohomology groups of the quantum torus using the stated complex I have found. I think there are ...
- 201
5
votes
0
answers
134
views
Projective dimension of ring over its center
If $A$ is a ring and $Z(A)$ is its center then what is a sufficient condition for the projective dimension of $A$ over $Z(A)$ (ie: $pd_{Z(A)}(A)$) to be finite?
(Assuming that $A\neq Z(A)$).
- 5,405
5
votes
0
answers
245
views
Orders of Clifford algebra
Let $C_n$ be the Clifford algebra over $\mathbb{Q}$ associated to negative definite quadratic form $-I_n$ (i.e. $-x_1^2-\dots-x_n^2$). Let $\mathcal{O}$ be a $\mathbb{Z}$-order of $C_n$.
Q1) Is it ...
- 1,692
5
votes
0
answers
137
views
Pseudovarieties of monoids
All (pseudo)varieties considered here are (pseudo)varieties of monoids.
It is known that any (finite or infinite) monoid that satisfies the identities
\begin{equation}
xhxyty = xhyxty, \quad xhytxy=...
- 563
5
votes
0
answers
295
views
Orbit-Stabilizer theorem for continuous groups
The orbit-stabilizer relationship (also known as the orbit-stabilizer theorem) is very clear for finite groups. Is there an equivalent relation for continuous groups?
Also, is there a similar notion ...
- 615
5
votes
0
answers
843
views
Chinese remainder theorem
For non-commutative rings, we have this generalization of the Chinese remainder theorem (CRT).
I wonder if there is another statement involving only left or right ideals;
do you know any?
- 59
5
votes
0
answers
246
views
Injectivity criterion for surjective coalgebra maps: does it hold in full generality?
Let $\mathbf{k}$ be a commutative ring.
Let $C$ be a filtered $\mathbf{k}$-coalgebra. This means a $\mathbf{k}$-coalgebra equipped with an increasing $\mathbf{k}$-module filtration $C^0 \subseteq C^1 \...
- 33.8k
5
votes
0
answers
578
views
Central Element in Sklyanin Algebras?
I'm interested in Sklyanin Algebras or Artin-Shelter regular algebras of type A. These are generated in degree 1 by three variables x,y,z, and have three defining relations in degree 2, which you can ...
- 807
5
votes
0
answers
442
views
A reference on semisimple linear algebra
Is there any literature where the tools familiar from (multi)linear algebra are systematically transferred to the setting of semisimple modules over noncommutative rings?
In fact this question is a ...
- 4,010
5
votes
0
answers
918
views
Commutator Baker-Campbell-Hausdorff formula
Consider the Baker-Campbell-Hausdorff formula $\Phi(X,Y)\in\mathbb{Q}\langle\!\langle X,Y\rangle\!\rangle$ in non-commutative variables. Define $X*Y:=\Phi(X,Y)$ and $[X,Y]=(-X)*(-Y)*X*Y$, and then (as ...
- 657
5
votes
0
answers
2k
views
Is the radical of a homogeneous ideal homogeneous?
Let $S$ be an $M$-graded $R$-algebra, where $M$ is some monoid, and $I\subset S$ an homogeneous ideal. The original, naïve, question, was: is it true that $\sqrt{I}$ is homogeneous? In this generality,...
- 1,811
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
5
votes
1
answer
226
views
Examples of noncommutative Bezout domains
I would like to see some (or many!) examples of noncommutative Bezout domains (one-sided principal ideals sum to one-sided principal ideals). I've read somewhere that it's not easy to find an example ...
4
votes
4
answers
1k
views
Why do we choose the standard total order on the integers?
I understand why the set of natural numbers $\mathbb N = \{ 0, 1, 2, \cdots \}$ is equipped with a total order. Indeed, every monoid has a pre-order, where $$n' \succeq n \quad \mathrm{if~and~only~if} ...
- 8,512
4
votes
4
answers
961
views
Homological dimension of a graded ring which is like polynomial ring
Let $k$ be a field of characteristic $0$. Consider the following $k$-algebra $R$, which is the quotient of a tensor algebra generated by elements $x_i$ in degree $1$ with the relation $x_ix_j=-x_jx_i$...
- 3,758
4
votes
3
answers
3k
views
Finitely generated projective = finitely presented flat over a noncommutative Noetherian ring
Let $R$ be a possibly noncommutative left Noetherian ring and $M$ an $R$-module. I am looking for a reference or a proof for the following fact: $M$ is finitely generated and projective if and only if ...
- 1,103
4
votes
2
answers
1k
views
Semiring naturally associated to any monoid?
For any monoid $M$, we can naturally construct a semiring $S$ as follows:
Let the additive monoid of $S$ be the free commutative monoid on $M$
Let the multiplicative monoid of $S$ be $M$
Then, if ...
- 4,907
4
votes
3
answers
567
views
Gröbner/SAGBI bases for non-commutative setting
It is well known that SAGBI/Gröbner bases are important for commutative and non-commutative algebra. The references for commutative scenery is ample and vast, but I am in trouble to find a good ...
- 829
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
4
votes
2
answers
586
views
Brauer group of $\mathbb{Z}_{(p)}$
This may be a well known result but I could not find it in the standard references. What is the Brauer group of the local ring $\mathbb{Z}_{(p)}$ (the ring of integers localized at $p$)?
- 81
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 ...
- 1,284
4
votes
2
answers
1k
views
strong nilpotent elements
An element x in a noncommutative ring R is strongly nilpotent if any chain
$x_1=x, x_2, ... $, with $x_{n+1}\in x_n R x_n$ terminates at zero. It becomes clear
why this is a good definition once one ...
- 43
4
votes
2
answers
544
views
Membership problem in monoids
What is the simplest example of a monoid with undecidable membership problem? In other words, I'm looking for a concrete monoid $S$ such that there is no algorithm which takes elements $s_1,...,s_n$ ...
- 41
4
votes
1
answer
222
views
Addition and Rudin-Keisler ordering in $\beta \omega$
$\DeclareMathOperator{\RK}{\mathrm{RK}}$Let $\beta\omega$ be the Stone-Cech compactification of $\omega$ with the discrete topology. We can endow $\beta\omega$ with an addition operation that extends ...
- 48.9k
4
votes
2
answers
1k
views
Semiprime (but not prime) ring whose center is a domain
The center of a prime ring is a domain and the center of a semiprime ring is reduced.
Now I have no evidence to believe that if the center of a semiprime ring R is a domain,
then R has to be a ...
- 279
4
votes
1
answer
270
views
Kaplansky inverse element theorem on group C-star algebra
In a class talking about $C^*$ algebra and (higher) index theory, I heard a theorem
(related to Kaplansky, proved?), that is
Suppose $\Gamma$ is a group (admitting Haar measure if necessary) while $\...
- 193
4
votes
3
answers
1k
views
Is there a general notion of semigroup action?
The analogous result to Cayley’s theorem or Yoneda lemma in semigroup theory represents semigroups as semigroups of functions from a set to itself. This suggests that a semigroup action consists of a ...
- 2,464
4
votes
1
answer
423
views
What is the formula for the commutative multiplication on CP(infinity)?
There is a classic formula for maps $\mathrm{CP}(r) \times \mathrm{CP}(s) \to \mathrm{CP}(r+s)$ or maybe $r+s+1$ using Plücker coordinates - IF memory serves. In the limit we get the abelian ...
- 3,880
4
votes
1
answer
239
views
True or false? Every left or right cancellative, duo semigroup is cancellative
A semigroup $S$ is duo if $aS = Sa$ for all $a \in S$, where $aS := \{ax: x \in S\}$ and similarly for $Sa$; for instance, every commutative semigroup is duo, and so is every group. On the other hand, ...
- 10.5k
4
votes
1
answer
196
views
Is the monoid of taking iterated images and inverse images freely generated by the image and inverse image operation?
Let $\mathcal{F}$ denote the class of all functions. Let $U,L:\mathcal{F}\rightarrow\mathcal{F}$ denote the mappings where if $f:X\rightarrow Y$, then $U(f):P(X)\rightarrow P(Y),L(f):P(Y)\rightarrow P(...
4
votes
1
answer
314
views
A semigroup with the property that $x^n = a$ has at least one solution
Is there a standard name for a (multiplicatively-written) semigroup $(A, \cdot)$ such that, given an arbitrary $a \in A$, the equation $x^n = a$ has at least one solution $x \in A$ for each $n \in \...
- 10.5k
4
votes
1
answer
151
views
Three preprints and one manuscript of Tamura on power semigroups
I'm reading Takayuki Tamura's article "On the recent results in the study of power semigroups", pp. 191-200 in Goberstein & Higgins' Semigroups and Their Applications, Kluwer, 1987 (the ...
- 10.5k
4
votes
1
answer
146
views
When is semigroup algebra local?
Let $G$ be a finite semigroup (or monoid if that helps) and $K$ a field.
Question: When is the semigroup algebra $KG$ local?
Here local means that there is a unique maximal right (or left) ideal.
...
- 26.5k
4
votes
1
answer
178
views
Can all finite-dimensional noncommutative algebras with trace be trace-preserving embedded into matrix rings?
Suppose I have a finite (non-)commutative ring $R/k$ (over a field $k$ of char $0$) with a linear "trace" function $t: R \to k$. Can I always find an embedding $f: R \to M_r(k)$ compatible ...
- 7,746
4
votes
1
answer
307
views
Characterization of Archimedean linearly ordered monoids
In this question, it is shown that all Archimedean ordered groups are isomorphic to an ordered subgroup of $\mathbb R$. Additionally, it is shown that if such a group is complete, then it is ...
- 249
4
votes
1
answer
243
views
number of indecomposable summands of an extension of two modules
I have the following question : in a Krull-Schmidt category (say the category of finite length left modules over a ring, this is the case which interests me), is it possible to relate the number of ...
- 333
4
votes
1
answer
189
views
Some questions about homogroups
Every semigroup containing an ideal subgroup is called a homogroup. Let $(S,\cdot)$ be homomgroup, hence it contains an ideal $I$ that is also a subgroup. It is easy to see that $I$ is the least ideal,...
- 995
4
votes
1
answer
297
views
Reference for subsemigroups of $\mathbb{N}^n$
A well known result about the natural numbers $\mathbb{N}$ says that for any finite subset $A \subset \mathbb{N}$ there exists $R \ge 0$ such that if $n$ is in the subgroup of $\mathbb{Z}$ generated ...
- 15.4k
4
votes
1
answer
434
views
Regarding a new algebraic structure
By "left semigroup-joined-semigroup" I mean an algebraic structures $(S,\cdot,*)$ such that both $\cdot,*$ are associative, and the following property holds (see this )
$$
x*(y\cdot z)=x*y*z\;\; ; \;...
- 995
4
votes
1
answer
855
views
Left- and right-sided principal ideals of quaternions have same index?
One fact about the Lipschitz integers (quaternions of the form $a + bi + cj + dk$ where $a, b, c, d$ are integers) is that the left-sided ideal generated by any element $Q$ has the same index in the ...
- 5,827
4
votes
1
answer
454
views
The Jordan Plane and Enveloping Algebras
Let $k$ denote a field of characteristic $0$ (assume algebraically closed for convenience). Define $J=k\langle x,y|[x,y]=y^{2}\rangle$. This noncommutative algebra (which can be viewed as a derivation ...
- 51