Questions tagged [noncommutative-algebra]

Non-commutative rings and algebras, non-associative algebras. Can be used in combination with ra.rings-and-algebras

Filter by
Sorted by
Tagged with
9 votes
1 answer
791 views

Combinatorics for the action of Virasoro / Kac–Schwarz operators: partition polynomials of free probability theory

In the background sections below, I establish the relations among characterizations of the action of Virasoro / Kac–Schwarz operators of 2D gravity models presented in terms of Laurent series by ...
Tom Copeland's user avatar
  • 9,937
7 votes
0 answers
563 views

Guises of the noncrossing partitions (NCPs)

From "Noncrossing partitions in surprising locations" by Jon McCammond: Certain mathematical structures make a habit of reoccuring in the most diverse list of settings. Some obvious ...
Tom Copeland's user avatar
  • 9,937
61 votes
3 answers
7k views

What is the current status of the Kaplansky zero-divisor conjecture for group rings?

Let $K$ be a field and $G$ a group. The so called zero-divisor conjecture for group rings asserts that the group ring $K[G]$ is a domain if and only if $G$ is a torsion-free group. A couple of good ...
Johan Öinert's user avatar
113 votes
2 answers
12k views

How would you solve this tantalizing Halmos problem?

$1-ab$ invertible $\implies$ $1-ba$ invertible has a slick power series "proof" as below, where Halmos asks for an explanation of why this tantalizing derivation succeeds. Do you know one? Geometric ...
Bill Dubuque's user avatar
  • 4,706
39 votes
9 answers
10k views

Simplest examples of rings that are not isomorphic to their opposites

What are the simplest examples of rings that are not isomorphic to their opposite rings? Is there a science to constructing them? The only simple example known to me: In Jacobson's Basic Algebra (...
Amritanshu Prasad's user avatar
17 votes
1 answer
1k views

Non-commutative Galois theory

Recall that an finite-dimensional algebra $A$ over a field $k$ is central simple iff there is an iso $A \otimes_k A^{op} \cong M_n(k)$ where $A^{op}$ is the opposite ring and $M_n(k)$ is the matrix ...
Jakob's user avatar
  • 1,986
16 votes
0 answers
845 views

Is "being a full ring of quotients" a Morita invariant property?

Definition and context: An (associative, unital, not necessarily commutative) ring $R$ is called classical if every regular element of $R$ is a unit. Equivalently, $R$ is its own classical ring of ...
Torsten Schoeneberg's user avatar
13 votes
3 answers
1k views

Are the trace relations among matrices generated by cyclic permutations?

Let $X_1,\dots,X_n$ be non commutative variables such that $\operatorname{tr} f(X_1,\dots,X_n) = 0$ whenever the $X_i$ are specialized to square matrices in $M_r(k)$ for any $r \geq 1$. Does this ...
Asvin's user avatar
  • 7,648
6 votes
3 answers
1k views

Are epimorphisms from a division ring isomorphisms ?

According to Corollary 1.2(3) of the paper Silver: Noncommutative Localizations and Applications. J. of Alg. 7(1964), 44-67: If $R$ is a (commutative) field and $\alpha: R \to S$ an epimorphism in ...
tj_'s user avatar
  • 2,170
6 votes
1 answer
235 views

Is there any structural characterization of the rings in which every element other than the identity is a (two-sided) zero divisor?

[I fear that I'm missing something obvious here, but I'll dare to ask anyway.] As we all know, a division ring is a (unital, associative, non-zero) ring where every non-zero element is a unit. So, let ...
Salvo Tringali's user avatar
42 votes
8 answers
2k views

How to quantify noncommutativity?

If I have two operators or finite-dimensional matrices $A$ and $B$, how can I quantify the amount to which they commute or don't commute? (I would consider it a big plus if it is computable easily for ...
Jiahao Chen's user avatar
  • 1,870
30 votes
2 answers
1k views

How to make the Capelli's identity less mysterious?

The formulation of the Capelli's identity is very elementary; it has important applications in invariant theory and representation theory, see http://en.wikipedia.org/wiki/Capelli%27s_identity To ...
asv's user avatar
  • 21.1k
29 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 ...
darij grinberg's user avatar
27 votes
5 answers
9k views

Can a quotient ring R/J ever be flat over R?

If $R$ is a ring and $J\subset R$ is an ideal, can $R/J$ ever be a flat $R$-module? For algebraic geometers, the question is "can a closed immersion ever be flat?" The answer is yes: take $J=...
Anton Geraschenko's user avatar
21 votes
1 answer
1k views

Is there any non-commutative ring such that every element other than the identity is a zero divisor?

A (unital) ring $R$ with the property that every element other than the identity $1_R$ is a (two-sided) zero divisor, seems to be commonly called a "$0$-ring" or "$\mathcal O$-ring"...
Salvo Tringali's user avatar
15 votes
2 answers
2k views

Why is "naive" definition of non-commutative spectrum bad?

It is well-known that the category of affine schemes is equivalent to the opposit category of commutative unital rings. So naively, one would think that the same should hold in non-commutative setting....
Sasha Patotski's user avatar
14 votes
9 answers
2k views

Examples of noncommutative analogs outside operator algebras?

Theo's question made me wonder if there are other "noncommutative analogs" outside of operator algebras. Some noncommutative analogs from operator algebras include: A $C^\ast$-algebra is a ...
Dave Penneys's user avatar
  • 5,335
14 votes
4 answers
2k views

Applications of Govorov-Lazard Theorem?

I asked this question on SE a long time ago, but never received an answer: The Govorov-Lazard Theorem states that a (left) module over an unital ring is flat iff it is a direct limit of finitely ...
Ralph's user avatar
  • 16.1k
13 votes
5 answers
3k views

Noncommutative localization of a ring: complete construction

I've been looking for the following construction in the literature, but I've only been able to find (very) partial proofs or proofs of special cases. Let $R$ be a non-commutative ring and $S$ a ...
Steve B's user avatar
  • 365
11 votes
1 answer
714 views

Determinants of octonionic hermitian matrices

For quaternionic hermitian matrices (i.e. quaternionic square matrices $(a_{ij})$ satisfying $a_{ji}=\bar a_{ij}$) there is a nice notion of (Moore) determinant which can be defined as follows. ...
asv's user avatar
  • 21.1k
10 votes
4 answers
2k views

Strongly Noetherian property. When is the tensor $A\otimes_{k}B$ Noetherian for Noetherian rings $A$ and $B$?

Let $k$ be a field. It is well-known that $A\otimes_{k}B$ is not necessarily Noetherian even if $k$-algebras $A$ and $B$ are Noetherian. For example $\mathbb{R}\otimes_{\mathbb{Q}}\mathbb{R}$. When ...
user2013's user avatar
  • 1,653
10 votes
1 answer
1k views

Explicit isomorphism for quaternion algebras over $\mathbb{Q}$?

It is known that the isomorphism class of a quaternion algebra $A=\binom{a,b}{K}$ over a number field $K$ is determined by the finite set of places $v$ of $K$ where $A\otimes_K K_v$ is a division ...
benblumsmith's user avatar
  • 2,831
9 votes
1 answer
498 views

Point modules of quantum projective space $\mathbb{P}^n$

Let $A$ be a quantum $\mathbb{P}^n$ defined by $$ A=\mathbb{C}\langle x_1,x_2,\dots,x_{n+1}\rangle/(x_ix_j-r_{ij}x_jx_i)_{1\le i < j\le n+1}. $$ I would like to know the set $X$ of isomorphism ...
user2013's user avatar
  • 1,653
8 votes
1 answer
428 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. ...
Chris Schommer-Pries's user avatar
7 votes
2 answers
2k views

Reason to apply the Koszul sign rule everywhere in graded contexts

The Koszul sign rule is a sign rule that arises from graded-commutative algebras. For instance, let $\bigwedge(x_1,\dots, x_n)$ be the free graded-commutative algebra generated by $n$ elements of ...
Javi's user avatar
  • 489
7 votes
3 answers
2k views

Units in a group algebra

Let k be a field and let G be a finite group. I would like to know if there is any nice description of the group of units in the group algebra kG. (If there is no nice answer in this generality, ...
Chebolu's user avatar
  • 575
6 votes
1 answer
1k views

Hochschild homology of quiver algebras

Let $K$ be a field and $\Gamma$ a quiver (=multidigraph) and $K[\Gamma]$ its quiver algebra (free $K$-module on the set of all paths of length $\geq0$ where multiplication is concatenation if ...
Leo's user avatar
  • 1,579
6 votes
2 answers
537 views

Properties of ring epimorphisms that are true only over commutative rings

I'm interested in knowing/collecting some properties of epimorphisms of rings (with identity) that are true over commutative rings but are false in the non-commutative case. Example: I learned from ...
tj_'s user avatar
  • 2,170
6 votes
1 answer
492 views

Do you know which is the minimal local ring that is not isomorphic to its opposite?

The most popular examples are non-local rings and minimal has 16 elements. I am interested in knowing examples of local rings not isomorphic to their opposite.
José María Grau Ribas's user avatar
6 votes
1 answer
360 views

Is every (left) graded-Noetherian graded ring (left) Noetherian?

I call a $\mathbb{Z}$-graded (non-commutative, associative, unital) ring $A$ (left) graded-Noetherian if every homogeneous (left) ideal is finitely generated, and (left) Noetherian if it is (left) ...
Anonymous Coward's user avatar
5 votes
0 answers
202 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 ...
Salvo Tringali's user avatar
5 votes
0 answers
193 views

A non-commutative analog of a result concerning a Jacobian pair

Let $k$ be a field of characteristic zero and let $E=E(x,y) \in k[x,y]$. Define $t_x(E)$ to be the maximum among $0$ and the $x$-degree of $E(x,0)$. Similarly, define $t_y(E)$ to be the maximum among $...
user237522's user avatar
  • 2,783
5 votes
1 answer
872 views

Why Jacobson, but not the left (right) maximals individually?

I firstly asked the following question on MathStackExchange a couple of months ago. I did not receive any answers, but a short comment. So, I decided to post it here, hoping to receive answers from ...
Kaveh's user avatar
  • 483
5 votes
0 answers
232 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 ...
Subhajit Jana's user avatar
5 votes
2 answers
833 views

Epimorphisms and free submodules

By inspecting the accepted answer to this question Are epimorphisms from a division ring isomorphisms ? one obtains the following necessary condition for epimorphisms: Let $R \le S$ be rings ...
tj_'s user avatar
  • 2,170
4 votes
2 answers
521 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 $p$)?
user123's user avatar
  • 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 ...
Fiktor's user avatar
  • 1,264
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 ...
Roman 's user avatar
  • 43
4 votes
1 answer
522 views

A Question on Koszul duality and $B(\infty)$ structures on $HH^*$

The following theorem is known from a paper "Duality in Gerstenhaber Algebras" by Felix, Menichi, Thomas. Given a simply connected space X of finite type. There is an equivalence of Gerstenhaber ...
Daniel Pomerleano's user avatar
4 votes
0 answers
150 views

Nilpotent operator of the Weyl algebra

For a research project I'm currently working on, I came across the following problem: Let $A=$ $^{k <x,y> }\Big/_{(yx-xy-1)}$ be the Weyl Algebra over a field $k$ of characteristic $p$, where $...
u1571372's user avatar
  • 489
3 votes
0 answers
227 views

Hurwitz–Radon problem for $ \mathbb{Q} ^{n} $

What is the maximal number of orthogonal operators $ A _{1} , \dotsc, A _ {m} $ in $ \mathbb{Q}^{ n } $ satisfying the relations $ A_{i}^{2} = - I $ and $ A_{i}A_{j} + A_{j}A_{i} = 0 $ for $ i \neq ...
Sky's user avatar
  • 913
3 votes
0 answers
386 views

Software for Combinatorial Algebra sought

I am looking for software which helps me do straightforward tasks in combinatorial algebra. Let me give an example of what I mean by a straightforward task: I have two graded (generally ...
3 votes
0 answers
140 views

Extension of work by Novelli and Thibon on noncommutative symmetric functions and Lagrange inversion

(Edit May 12, 2023: I just put up a brief summary of some of my notes on the partition polynomials described below in my WordPress mini-arXiv at "As Above, So Below". It contains multinomial ...
Tom Copeland's user avatar
  • 9,937
3 votes
1 answer
164 views

$M$ is finitely generated as $A$-module iff $M/A_{>0}M$ is finitely generated as $A$ module?

Let $A$ be a nonnegatively graded algebra and $M$ a nonnegatively graded $A$-module. Then, $A_{>0}M$ is a graded $A$-submodule of $M$. (Here, given a nonnegatively graded algebra $A$, we've ...
Jakob W's user avatar
  • 339
3 votes
0 answers
92 views

Is a specific endomorphism of $A_1$ an automorphism?

Let $k$ be a field of characteristic zero, and let $A_1(k)$ be the first Weyl algebra, namely, the associative non-commutative $k$-algebra generated by $x$ and $y$ subject to the relation $yx-xy=1$. ...
user237522's user avatar
  • 2,783
3 votes
1 answer
364 views

Dimension of hermitian rank at most $k$ matrices over quaternions

In a (right) finite dimensional quaternionic Hilbert space there is an analogue of the spectral theorem (see theorem 4.6 in Farenick and Pidkowich) for normal matrices in $\mathbb{H}^{m\times m}$, ...
Josiah Park's user avatar
  • 3,177
2 votes
1 answer
186 views

Origins of a theorem on an atomic factorizations in domains and cancellative monoids satisfying the ACCPL and the ACCPR

Let $H$ be a (commutative or non-commutative) monoid. We say that $H$ satisfies the ACCPL (ascending chain condition on principal left ideals) if there exists no infinite sequence of principal left ...
Salvo Tringali's user avatar
2 votes
1 answer
430 views

dg-resolution of the polynomial algebra

I am intersted in constructing a cofibrant resolution of the commutative polynomial algebra in some number of variables in the category of dg-algebras(not necceserily commutative). The resolutions ...
lks8271's user avatar
  • 165
2 votes
0 answers
116 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 ...
wlad's user avatar
  • 4,823
2 votes
1 answer
439 views

Algebra of endomorphisms of f.g. modules as subquotients of matrix algebras

Let $A$ be a $C$-algebra, where $C$ is a commutative ring with $1$, and $M$ be a finitely generated left $A$-module. Question: Is it true that we can always find a positive integer $n$, a $C$-...
carlos's user avatar
  • 279