All Questions
Tagged with noncommutative-algebra noncommutative-rings
92 questions
5
votes
1
answer
883
views
Is this ring isomorphic to a quotient of a group algebra?
Consider the quotient of the free algebra $\mathbb{Q}\langle \alpha, \beta, \gamma, \delta, \varepsilon, \zeta \rangle$ by the two-sided ideal $I$ subject to the relations $$ \alpha\delta=\delta\alpha=...
- 1,093
4
votes
0
answers
131
views
Is there anything like a Čech complex for calculating local cohomology over *noncommutative* rings?
$\DeclareMathOperator\Mod{Mod}\DeclareMathOperator\colim{colim}$Let $R$ be a ring, and consider a two-sided ideal $I = (r_1, r_2, \dots, r_j)$ in $R$. The corresponding $n$th local cohomology functor ...
- 1,335
7
votes
1
answer
653
views
Which CAS can do basic non-commutative differential algebra?
This is a repost of my question at MSE from 7 months ago, to which I haven't been able to find an answer yet.
I am looking for a CAS (possibly incl. additional packages/libraries) that can compute ...
- 7,127
1
vote
0
answers
69
views
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
10
votes
1
answer
221
views
Matrix ring isomorphisms of different sizes
Do there exist (unital, associative, noncommutative) rings $R$ and $S$, where $\mathbb{M}_2(R)\cong \mathbb{M}_3(S)$, but these matrix rings are not isomorphic to $\mathbb{M}_6(T)$ for any ring $T$?
- 18.7k
1
vote
1
answer
368
views
Wedderburn theorem for finite-dimensional algebras over the complex numbers
I'm trying to understand how to apply the Wedderburn theorem in the context of unitary algebras over $\mathbb{C}$ that are finite-dimensional and semisimple. Let $\mathcal{A}$ be a $\mathbb{C}$-...
2
votes
2
answers
226
views
Minimal ideals and subalgebras of semisimple algebras
I'm considering an algebra to be a ring which is also a vector space over some field $F$, and the algebra $A$ is said to be semisimple if it is semisimple as a ring, i.e., $A$ can be written as a ...
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
2
votes
0
answers
144
views
Zero divisors in the extra-special group algebra $\mathbb{R}[2^{1+6}_+]$
Can you characterize the unit-group of the real group-algebra of the extraspecial plus-type 2-group of order 128? (That is $\mathbb{R}[2_+^{1+6}]$ using Conway's notation.)
(Please choose any irrep ...
- 21
4
votes
0
answers
80
views
Indecomposable injectives over Weyl algebras
Let $A=A_n(\mathbb{C})$ be the $n$-th Weyl algebra over the complex field. Then $A$ is a left Noetherian noncommutative ring. Is there a complete classification of indecomposable injective $A$-modules?...
- 615
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
1
answer
165
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Polynomial identities satisfied by the Weyl algebra in prime characteristic
The rank $n$ Weyl $A_n(\mathsf{k})$ algebra over a field $\mathsf{k}$ of zero characteristic does not satisfies any polinomial identity. If it were a PI-algebra, Kaplansky theorem would apply (since ...
- 3,318
1
vote
0
answers
108
views
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 $n$-...
- 331
2
votes
1
answer
181
views
Relation(s) between units and nilpotent elements in graded noncommutative rings
In Commutative Algebra we have the following standard facts which I am going to state in a slightly different form than usually found in textbooks. Namely, let $A$ be a commutative unital ring of ...
- 7,127
3
votes
1
answer
375
views
Graded global dimension of a graded algebra
Let $k$ be an algebraically closed field of characteristic $0$.
Let $A := k \langle x,x^{-1},y \rangle /(xy-qyx, x^{d_1}-ay^{d_2})$, where deg$(x)>0$, deg$(y)>0$, $q,a \in k^*$ and $d_1\text{deg}...
- 889
2
votes
0
answers
172
views
Simple modules of quantum planes
Let $k$ be an algebraically closed field.
Let $R := k\langle x,y \rangle/(yx-qxy) (q \in k^*)$.
We often call $R$ a quantum plane.
If $q$ is a primitive $n$-th root, then for any $(\zeta, \xi) \in k^* ...
- 889
6
votes
0
answers
255
views
Proving the spectrum of the Young-Jucys-Murphy elements by formal computation in the degenerate affine Hecke algebra
This is really a followup to Why are Jucys-Murphy elements' eigenvalues whole numbers? , specifically to Igor Makhlin's beautiful answer. I'm trying to make it even more beautiful by getting rid ...
- 33.8k
4
votes
0
answers
91
views
Nullstellensatz for maximal left ideals of quantum plane
Let $R=\mathbb{C}\langle x,y\rangle/\langle xy=qyx\rangle$ be the quantum plane algebra. Does some sort of Nullstellensatz holds for the maximal left ideals of $R$? By this we mean all maximal left ...
- 91
1
vote
0
answers
156
views
Pseudo-coherent complexes over sheaves of non-commutative rings
I am posing a question on derived categories to which I was not able to find an answer anywhere in the literature. I would appreciate any answer, hint or suggestion.
Assume that $\mathcal{R}_X$ is a ...
- 609
3
votes
0
answers
88
views
Explicit separability idempotent for the center of a separable algebra
Let $A$ be a $k$-algebra for some commutative ring $k$. Recall that $A$ is said to be separable over $k$ if the multiplication map $A\otimes_k A^{\operatorname{op}}\to A$ has a section as a map of $A\...
- 15.8k
4
votes
0
answers
166
views
A non-commutative, left duo ring whose only unit is the identity
Let $R$ be a ring (here, rings are always associative, unital, and non-zero). We say that $R$ is a left duo ring if $aR \subseteq Ra$ for every $a \in R$.
Question. Is there a non-commutative, left ...
- 10.5k
5
votes
1
answer
266
views
Rings s.t. each element has a power lying in the center (and their completely prime ideals)
Let $R$ be a ring (throughout, all rings are associative and unital). We say $R$ satisfies condition (C) if, for every $a \in R$, there exists an integer $n \ge 1$ (depending on $a$) such that $a^n$ ...
- 10.5k
2
votes
2
answers
181
views
Module complements to rings embedded in a projective module
Let $R$ be noncommutative unital ring and $M$ a projective (right) $M$-module. Assume that $R$ embedds into $M$ as a right -module.
A) If $R$ is a semisimple ring, then of course $R$ admits an $R$-...
- 203
2
votes
1
answer
287
views
Every module of finite uniform dimension is a direct sum of (finitely many) indecomposable
Crossposted on StackExchange on July 28 (no answer so far).
Let $R$ be a (commutative or non-commutative, associative, unital) ring. It is well known that any artinian or noetherian $R$-module $M$ can ...
- 10.5k
26
votes
3
answers
724
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Subtraction-free identities that hold for rings but not for semirings?
Here is a concrete, if seemingly unmotivated, aspect of the question I am interested in:
Question 1. Let $a$ and $b$ be two elements of a (noncommutative) semiring $R$ such that $1+a^3$ and $1+b^3$ ...
- 33.8k
2
votes
0
answers
98
views
Do $r(a) \leq^\oplus R$ and $r(a) = r(a^2)$ imply $r(a) = eR$ and $aR \subseteq (1-e)R$ for some idempotent $e$?
Let $R$ be a (commutative or non-commutative, associative) ring with unity, and let $a$ be an element of $R$ such that $r(a) = r(a^2)$, where $r(\cdot)$ denotes a right annihilator. It follows that $r(...
- 10.5k
1
vote
0
answers
39
views
Rings where every indecomposable principal right ideal is extensive
Let $R$ be a (commutative or non-commutative, associative) unital ring. Following Nicholson & Yousif [1, p. 21], we say that a right ideal $\mathfrak i$ of $R$ is extensive if every $R$-linear ...
- 10.5k
2
votes
2
answers
105
views
Every non-zero submodule of $R_R$ has an indecomposable direct summand: True when $R$ is von Neumann regular?
Let's say that a (right) module $M$ is well complemented if every non-zero submodule of $M$ has an indecomposable direct summand (by the way, is there a better or more standard name for this property?)...
- 10.5k
1
vote
0
answers
124
views
On the rings $R$ with the property that $eR \cong fR$ for all primitive idempotents $e, f \in R$
Let $R$ be a (commutative or non-commutative) ring with identity. As usual, an idempotent $e \in R$ is primitive if $eR$ (the principal right ideal generated by $e$) is indecomposable as a right ...
- 10.5k
2
votes
1
answer
192
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 ...
- 10.5k
21
votes
1
answer
2k
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"...
- 10.5k
6
votes
1
answer
256
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 ...
- 10.5k
1
vote
1
answer
402
views
Are there algebras over reals besides complex numbers, where identities, analoguous to $(-1)^i=e^{-\pi}$ and $i^i=e^{-\pi/2}$ hold?
Are there algebras over real numbers (with exponentiation), such that there is such $z$ that does not include components in $\mathbb{C}$ (or in a subset isomorphic to $\mathbb{C}$), for which $(-1)^z\...
- 10.1k
2
votes
1
answer
107
views
The ring of upper triangular $n$-by-$n$ matrices over a skew field is (left and right) Rickart
Let $T_n(R)$ be the ring of upper triangular $n$-by-$n$ matrices with entries in a (commutative or non-commutative) unital ring $R$. It happened to me to note that, if $R$ is a skew field, then $T_2(R)...
- 10.5k
1
vote
1
answer
98
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.
...
- 10.5k
1
vote
1
answer
147
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
234
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;...
2
votes
0
answers
270
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 ...
- 5,230
6
votes
1
answer
499
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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.
0
votes
0
answers
43
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 $P \cap (R \otimes 1) = (0) = P \cap (1 \otimes S)$, ...
- 1
4
votes
1
answer
193
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 ...
- 248
0
votes
0
answers
96
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 ...
- 131
11
votes
3
answers
908
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. ...
- 5,230
6
votes
1
answer
372
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 ...
- 161
6
votes
2
answers
543
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 ...
- 10.5k
2
votes
0
answers
50
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 ...
- 63.9k
9
votes
1
answer
248
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 ...
- 63.9k
2
votes
0
answers
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
5
votes
1
answer
403
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 $R = \Lambda\{F\}$...
- 7,746
2
votes
0
answers
56
views
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