# Questions tagged [noncommutative-algebra]

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

393
questions

**1**

vote

**1**answer

61 views

### If, in a unit-regular ring, the right annihilator of $a$ equals the right annihilator of $b$, then $aR = bR$?

Recall that a (unital) ring $R$ is von Neumann regular (VNR) if, for each $x \in R$, there exists $y \in R$ such that $x = xyx$; and unit-regular if such an element $y$ can be taken to be a unit.
...

**-6**

votes

**0**answers

60 views

### non commutative algebra or operator theory [closed]

Is Hurwitz-Radon theorem is true for the field $ \mathbb{Q} $. Actually I want to know when $M_{n}(\mathbb{Q}) $has n- subspace of dimension n?
links-<PROBLEMS AND THEOREMS
IN LINEAR ALGEBRA,by V. ...

**2**

votes

**0**answers

75 views

### 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,...

**10**

votes

**2**answers

428 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 ...

**4**

votes

**1**answer

123 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 ...

**3**

votes

**1**answer

212 views

### Number of rings with additive group $(\mathbb{Z}_{16})^2$. A341547(16) in OEIS

I would like to know if somewhere the number of non-isomorphic rings with additive group $(\mathbb{Z}_{16})^2$ is mentioned. If not, is someone able to calculate it?
And (easier) the commutative case? ...

**4**

votes

**1**answer

92 views

### Hopf “algebroid” structure of a groupoid convolution algebra?

This question is already posted in math.stackexchange, but didn't receive any answer. I'm not sure if this question fits in here, but surely someone in here can guide me to the correct answer.
To make ...

**4**

votes

**0**answers

113 views

### Buchberger's criterion for Gröbner bases in $k$-algebras with multiplicative basis and admissible order

Let $R$ be an associative $k$-algebra with multiplicative basis $\mathcal B$ with an admissible order on $\mathcal B$.
Let $G \subseteq R$ be a subset.
A multiplicative basis $\mathcal B$ means that $...

**1**

vote

**1**answer

127 views

### Do you know a finite unitary reversible ring that is not isomorphic to its opposite? And the minimal with that property?

Do you know a finite unitary reversible ring that is not isomorphic to its opposite? And the minimal with that property?
The examples of rings not isomorphic to their opposite that I know of are not ...

**4**

votes

**0**answers

141 views

### Do you know rings without involutions, auto-anti-isomorphics? In that case, what is the minimal example?

Do you know rings without involutions, but auto-anti-isomorphic (isomorphic to their opposite)? In that case, what is the minimal example?
If a ring has an involution f, then f is an anti-automorphism;...

**6**

votes

**0**answers

102 views

### Expressing a polynomial as the determinant of a matrix of linear forms

I have heard that it's a well known result (in theoretical computer science?) that if we have a polynomial $p(t_1,\dots,t_n)$ over $\mathbb Q$, we can find matrices $M_0,\dots,M_n/\mathbb Q$ such that ...

**1**

vote

**0**answers

56 views

### Strongly graded rings

In Theorem 3.1 of Graded rings over arithmetical orders, the authors prove that for a strongly $\mathbb{Z}$-graded ring $R$, if $R_0$ is left and right Goldie and a maximal order in its (classical) ...

**3**

votes

**0**answers

115 views

### Rough paths, unparametrized path space, and Kontsevich's moduli space of stable maps

Let $X$ be a manifold. Modulo reparametrization, the path space of $X$ is a groupoid $\Pi_X$. In Kapranov's "Free Lie Algebroids and the Space of Paths", Kapranov constructs an associated ...

**1**

vote

**0**answers

42 views

### A list of low-dimensional associative $\mathbb{C}$-algebras with non-trivial centers

I am looking for a list of (examples of) low(est)-dimensional non-commutative associative unital $\mathbb{C}$-algebras $A$ with non-trivial centers $Z(A)$ (hence non-semisimple). For our purposes $Z(A)...

**1**

vote

**0**answers

33 views

### Separable nonassociative algebras

In his paper "Structure and Representation of Nonassociative Algebras", Schafer notes that an arbitrary nonassociative algebra over a field is separable "if and only if the ...

**1**

vote

**0**answers

40 views

### Maximal orders separable over their centre

Let $\mathcal{A}$ be a central simple $K$-algebra, where $K$ is an algebraic number field. It is known that $\mathcal{A}$ is separable over $K$ (following the definition of DeMeyer and Ingraham's book)...

**1**

vote

**0**answers

153 views

### 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 ...

**3**

votes

**2**answers

276 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 $p$)?

**3**

votes

**0**answers

69 views

### The type number of an algebra

I've been reading On the existence of maximal orders, by C.F. Yu, in which he discusses maximal $R$-orders in semisimple algebras over a field $K$, where $R$ is a Noetherian integral domain and $K = \...

**5**

votes

**1**answer

355 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.

**4**

votes

**0**answers

96 views

### non-abelian tensor products of several groups

R. Brown and J-L. Loday had defined the tensor product of two arbitrary groups acting on each other. Let $G,H$ be groups with actions on each other on the right. each group act on itself by ...

**5**

votes

**1**answer

86 views

### Integral monoid rings and Ore conditions

Consider a cancellative monoid $S$ satisfying the left Ore condition, so it embeds in a group $G=S^{-1}S$. Consider also the integral monoid rings $\mathbb Z[S]$ and $\mathbb Z[G]$.
I have two, ...

**2**

votes

**1**answer

154 views

### Top and bottom composition factors of $M$ are isomorphic

Let $k$ be a field and $N$ a finite group. Let $M$ be a projective indecomposable $kN$-module. Since the algebra $kN$ is symmetric, it follows that the top and bottom composition factors of $M$ are ...

**1**

vote

**0**answers

33 views

### Monomials in right ideal

First of all, hello everyone and thanks in advance of any kind of help.
Let $X =\{x_1,\dots,x_n\}$ and denote by $K\langle X\rangle$ the free algebra over some field $K$. Let $I = (f_1,\dots,f_m) \...

**4**

votes

**0**answers

45 views

### Color algebras and color involutions

If $A$ is a $G$-graded algebra then one can define on it a color involution, i.e. a bijective linear map preserving the grading such that the image of a product of two homogeneous elements is defined ...

**0**

votes

**0**answers

39 views

### Can Q(R) embed to Q((R ⊗ S )/ P)

Let $R, S$ be Noetherian $k$-algebra, where $k$ is a field, and $P \otimes S$ is Noetherian.
let $P$ be a prime ideal of $R \otimes S$ such that $P \cap (R \otimes 1) = (0) = P \cap (1 \otimes S)$, ...

**0**

votes

**0**answers

35 views

### Finding monomials in two-sided ideal of free algebra

Hello everyone and thanks a lot in advance for all your help!
I am currently working with a (finitely generated) two-sided ideal $I$ in the free algebra $K\langle X\rangle$ over a field $K$. I am ...

**5**

votes

**1**answer

140 views

### $R/I\cong R/\text{Ann}_R(R/I)$ but $I\neq\text{Ann}_R(R/I)$

I originally asked this on Stack Exchange but with no luck. Upon doing research with some noncommutative rings, I thought of a curious question. Does there exist a noncommutative unital ring $R$ and ...

**0**

votes

**0**answers

73 views

### Gelfand–Kirillov dimension of the first Weyl algebra by using the definition

$\DeclareMathOperator\GKdim{GKdim}$Here I am trying to find the Gelfand–Kirillov dimension of the first Weyl algebra just by using the definition of the Gelfand–Kirillov dimension.
Let $A$ be an ...

**7**

votes

**1**answer

214 views

### Weyl algebra as an Azumaya algebra over its centre

Assume that $k$ is an algebraically closed field of positive characteristic $p$. On page 3 (page 6 of the PDF file) of Bezrukavnikov, Mirković, and Rumynin - Localisation of Modules for a semisimple ...

**6**

votes

**1**answer

133 views

### Survey of recent developments of the Gelfand-Kirillov dimension

It is almost two decades since the now classical books by McConnell and Robinson's
[ Noncommutative Noetherian rings. With the cooperation of L. W. Small. Revised edition. Graduate Studies in ...

**3**

votes

**0**answers

62 views

### On noncommutative transcendence degrees

The original transcendence degree for (noncommutative) division algebras is the Gelfand-Kirillov transcendence degree, due to I. Gelfand and K. Kirillov ([ Sur les corps li´es aux algèbres ...

**11**

votes

**3**answers

744 views

### Does Morita theory hint higher modules for noncommutative ring?

Two possibly noncommutative rings are called Morita equivalent if their left-module categories are equivalent. In the commutative case, Morita equivalence is nothing more than ring isomorphism. ...

**3**

votes

**0**answers

86 views

### On the Gelfand-Kirillov Conjecture

The base field $k$ is of zero characteristic.
Notation: $A_{n,s}(k):= A_n(k(x_1,\ldots,x_s))$, the Weyl agebra over a purely transcedental extension of the base field; $F_{n,s}(k)$, the Weyl field, is ...

**6**

votes

**0**answers

127 views

### Differential birational equivalence

Suppose the base field algebraically closed and of zero characteristic.
There are two fascinating questions in the intersection of ring theory and algebraic geometry (for which an excellent discussion ...

**0**

votes

**3**answers

355 views

### non-associative but commutative algebra [closed]

Is it possible(or may be easier) to give an example of non associative algebra but commutative?
At first sight, it seems possible to prove associativity from commutativity but later realised it may no ...

**3**

votes

**1**answer

112 views

### Gelfand-Kirillov dimension of the first Weyl algebra

How can we compute the Gelfand-Kirillov dimension (GK for short) of the first Weyl algebra?
As we know we can look at the Weyl algebra as a generalized Weyl algebra in the following way:
Let $A=\...

**7**

votes

**1**answer

219 views

### Constructive definition of noncommutative rational functions (aka free skew fields)

The question
Let $F$ be a field. (I am fine with assuming $F=\mathbb{Q}$, but I suspect
that a "right" answer will be independent of $F$.) Let $k$ be a nonnegative integer.
Question. Is ...

**4**

votes

**1**answer

179 views

### Cohn localization examples

I'm working on my master's thesis, part of which involves an exposition on Cohn localization. (nlab discussion)
In Free ideal rings and localization in general rings, Sec 7.4, Cohn gives a ...

**6**

votes

**2**answers

260 views

### Rings $R$ such that every [regular] square matrix with entries in $R$ is equivalent to an upper triangular matrix

Let $\text{M}_n(R)$ be the ring of $n$-by-$n$ matrices with entries in a commutative unital ring $R$. Theorem III in
C.R. Yohe, Triangular and Diagonal Forms for Matrices over Commutative ...

**2**

votes

**0**answers

49 views

### 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 ...

**7**

votes

**1**answer

164 views

### Categories of modules generated under coproducts by a small set?

Question 1: For which rings $R$ does there exist a small set $S \subseteq Mod_R$ such that every module $M \in Mod_R$ is a direct sum of modules in $S$?
Equivalenty, for which rings $R$ does there ...

**2**

votes

**0**answers

92 views

### 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 ...

**3**

votes

**0**answers

66 views

### A graded version of the Koethe conjecture

The most straightforward graded version of the Koethe conjecture would seem to say that the result of summing of two graded-nil ideals produces a graded-nil ideal. Here, graded-nil means having all ...

**2**

votes

**0**answers

78 views

### 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 ...

**11**

votes

**0**answers

243 views

### Criteria for a map of rings to induce an equivalence on K-theory?

Algebraic $K$-theory is Morita invariant, but surely it does not detect Morita equivalence. What are some examples of rings (or ring spectra) $R$ and $S$ that are not Morita equivalent, but ...

**3**

votes

**1**answer

176 views

### Hochschild homology of acyclic complex

Let $A$ be a differential graded algebra over a commutative ring $R$. Suppose that $H_*(A)=0$, i.e. $A$ is acyclic.
Question: Does this imply that the Hochschild homology $HH_*(A)$ also vanishes ...

**5**

votes

**1**answer

208 views

### Classification of finitely generated modules over non-commutative rings

Let $\Lambda$ be a commutative integral ring with an automorphism $\sigma$ (I have in mind $\mathbb Z_p[[t]]$ and $\sigma(t) = (1+t)^\alpha - 1$ with $\alpha \in \Lambda^\times$) and $R = \Lambda\{F\}$...

**1**

vote

**0**answers

48 views

### Universal bimodule for homotopy biderivations

Recall the commutative story: for a commutative algebra $A$, its module of differentials $\Omega (A)$ is characterized by the universal property that any derivation $\delta \colon A \to M$ is in a ...

**3**

votes

**2**answers

644 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 ...