Questions tagged [noncommutative-algebra]
For questions about rings which are not necessarily commutative and modules over such rings.
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8 views
Is there a Grassman equivalent of orthogonal functions?
Orthogonal functions such as the Hermite functions work with commutative variables. Is there a similar thing that works with Grassman variables?
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1answer
42 views
Noncommutative rings and prime/maximal ideals
Let $R$ a non-simple noncommutative ring, and let $\mathcal{I}$ the set of non-trivial (right, left) ideals of $R$, with the following property: "Every element $I \in \mathcal{I}$ is prime and/or ...
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1answer
24 views
Computing the left annihilator $\text{Ann}_R(1_R-a)$ of a ring $R$.
I was computing the left annihilators of the elements $x$ of a ring $R$ with $1_R$ (denoted by Ann$_R(x)$) and encountered the following scenario:
For any $a\in R$, $$\text{Ann}_R(1_R-a)=\{r\in R: r(...
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1answer
45 views
Is it true to say that : Every submodule of a module M contains in a maximal submodule?
Let $R$ be an arbitrary ring with $1\neq 0$ and $_RM$ a left $R$-module. Is it true to say that : Every proper submodule of a module $M$ is contained in a maximal submodule?
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1answer
87 views
Determining the structure of a finite ring
I'm working on the following problem and would like some guidance
$R$ is a finite ring with $x^5 = x$ for every $x$ in $R$. Determine the structure of $R$.
My first thoughts were to factor and ...
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0answers
25 views
Definition of socle of a module
For me, the socle of an $R$-module $M$ is the unique maximal semisimple submodule of $M$. If $(R,\mathfrak m)$ is a local ring, this is equivalent to the maximal submodule annihilated by $\mathfrak m$....
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1answer
32 views
Semisimple subalgebras of $M_4(\mathbb{C})$
I'm working on the following problem and I'd like some guidance.
Describe up to isomorphism all semisimple $\mathbb{C}$-subalgebras of $M_4(\mathbb{C})$ (4 by 4 matrices over $\mathbb{C}$). Note ...
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1answer
28 views
Boson operator algebra - unitary transformation
What is the simple way to evaluate
$$ e^{i \alpha n(n-1)}a^{\dagger} e^{-i \alpha n(n-1)}$$
and
$$ e^{i \alpha n(n-1)}a e^{-i \alpha n(n-1)}$$
where $n=a^\dagger a$ and $a$ and $a^\dagger$ are ...
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0answers
21 views
Computations in matrix space over noncommutative semiring
Let $M^{n\times n}$ is a matrix space over a noncommutative idempotent finite semiring $S$. $X\in M,L\in M,R \in M$. Whether exists a way to compute a sequence of the form $((((((X*L)^T * R^T)^T * L)^...
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2answers
60 views
Complicated differential operator and application of the Zassenhaus formula
I have a differential operator
$$D_x = \frac{1}{x} \left[ (x^2 - a^2) \frac{d}{dx} \left( \frac{1}{x} \frac{d}{dx} \right) + 2 \frac{d }{dx} \right], $$
and I would like to compute
$$ e^{i \...
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1answer
28 views
Is $\operatorname{Hom}(E,M)$ simple as an $\operatorname{End}(M)$-module if $E$ is simple?
Let $R$ be a ring, let $M$ be an $R$-module and let $R' := \operatorname{End}_R(M)$.
Let $E$ be a simple $R$-module with $\operatorname{Hom}_R(E,M) \neq 0$.
Question:
Is $\operatorname{Hom}_R(E,M)...
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1answer
47 views
Matrix trace equality [closed]
Suppose $A$ and $B$ are positive definite matrices of the same size. Prove that
\begin{align}
\operatorname{Tr}( A^{1/2} (A^{-1/2} B A^{-1/2})^{1/2} A^{1/2}) \leq \operatorname{Tr}( (A^{1/2} B A^{1/2}...
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0answers
56 views
Is pullback of non-commutative rings well defined?
I know that pullback is defined for commutative ring, but what about non-commutative case?
Let's consider the following diagram, where $R_i,\bar{R}$ are rings and $R$ is the pullback:
Then $1\in R$ ...
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1answer
25 views
Constructing an integral domain with a specific subring
What tools are available for constructing various (noncommutative) domains with a given subdomain? That is, how can I begin to examine various domains which contain the domain $S$, other than ...
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0answers
52 views
Reference: Every Scheme is Derived Affine
I'm trying to track down a reference for the following claim, found in Kaledin's lectures Methods in Noncommutative Algebraic Geometry:
As it turns out, an arbitrary scheme $X$ also appears ...
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1answer
28 views
Is Artinian algebra with finite injective dimension of itself is self-injective？
Let $A$ be an Artinian $R$-algebra .(i.e.there is a ring homomorphism $\alpha:R\rightarrow A$,where $R$ is a commutative Artinian ring,the image of $\alpha$ lies in the center of $A$ and $A$ is ...
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0answers
46 views
Jacobson radical of the $\mathbb{k}$-algebra $\mathbb{k}[x]/\langle{p(x)}\rangle$ where $\mathbb{k}$ is a field
Let $\mathbb{k}$ be a field and $p(x)\in\mathbb{k}[x]$. I am trying to prove that the Jacobson radical (the intersection of all maximal ideals of $A$, denoted by $\mathrm{rad}(A)$) of the $\mathbb{k}$-...
3
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0answers
42 views
Lang's proof of the uniqueness of multiplicities for semisimple modules
In Algebra by Serge Lang (XVII, Proposition 1.2, p. 643-644) it is shown that for a simple $R$-module $E$ and $n, m \geq 0$ with $E^{\oplus n} \cong E^{\oplus m}$ it follows that $n = m$.
Lang argues ...
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0answers
32 views
Natural transformations from coinvariants to invariants
Consider a group $G$ and a ring $R$. Write $R[G]\overset{\varepsilon}{\longrightarrow}R$ for the counit map. Write $(-)_G\dashv \varepsilon ^\ast \dashv (-)^G$ for the adjoint triple on modules ...
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1answer
219 views
The mysterious isomorphism between coinduction and induction
Let $G$ be a finite group.
This answer mentions an explicit yet mysteriously preferable isomorphism from the coinduction functor to the induction functor.
This instructive answer observes the ...
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0answers
31 views
Is every (left) graded-Noetherian graded ring (left) Noetherian?
I call a graded (non-commutative) ring $A$ (left) graded-Noetherian if every homogeneous (left) ideal is finitely generated, and (left) Noetherian if it is (left) Noetherian as a ring.
In the ...
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0answers
24 views
How do I prove that every fractional ideal of an order in a division algebra is a full lattice?
Let O be an R-order for some Dedekind domain R, let F be the field of fractions of R and D be a division algebra over F. A fractional left ideal of O is an R-lattice I in D such that OI in I (I ...
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0answers
39 views
Are all quaternion algebras over the rationals skew fields?
If I understand correctly, any quaternion algebra over the rationals is a noncommutative associative division algebra. I am currently working with implementations of quaternion algebras in MAGMA and ...
3
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0answers
17 views
Does Pontryagin duality extend to class two nilpotent or maybe even metabelian locally compact groups?
A metabelian group $G$ is determined up to isomorphism by "abelian data" $(A,M,\alpha)$ -- an abelian group $A:=G/[G,G]$, an $A$-module $M:=[G,G]$, and a cocycle $\alpha:A\times A\to M$ giving the ...
2
votes
1answer
56 views
group-like elements of a Hopf algebra and the group algebra
Suppose we are given a $n$-dimensional Hopf algebra $H$ over field $\mathbb K$. I want to prove that
$H$ contains n distinct group-like elements if and only if there exists a group $G$ such ...
3
votes
2answers
65 views
Maximal ideals in a group ring when the ring is a field
Is there any simple characterization of the maximal ideals in a group ring $R[G]$ when $R$ is a field, perhaps in terms of maximal subgroups of $G$?
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2answers
32 views
Field where $e^2 + f^2 = -1$ implies existance of matrix $A$ with $A^2 = -E_2$?
Suppose we have a field $K$ with a solution for $e^2 + f^2 = -1$. Is there a matrix $A \in K^{2\times 2}$ with $$ A^2 = \begin{pmatrix}-1 & 0 \\ 0 & -1 \end{pmatrix}? $$
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28 views
Solving linear systems for integer values in MAGMA
Say we are given a quaternion algebra D over a number field F as well as a maximal $\mathcal{O}_F$-order $\Delta$ $\subseteq$ D and say we have a $\mathbb{Z}$-basis $\omega_1, . . . , \omega_n$ for $\...
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1answer
29 views
How to show that an ideal I of a ring R is semiprimitive if and only if I is an intersection of primitive ideals
A semiprimitive ideal in a ring R is any ideal I such that $J(R/I) = 0$.
I know that for any ring R, the Jacobson radical, $J(R)$, is the intersection of all the left (resp., right) primitive ideals ...
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0answers
30 views
Computing the inverse of a full lattice in a quaternion algebra
Let $D$ be quaternion algebra over a number field $F$. Let $\Delta\subseteq D$ be a maximal $\mathcal{O}_{F}$-order. Let $\mathfrak{b}$ be a fractional left $\Delta$-ideal. In his book "Maximal Orders"...
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0answers
29 views
What is a relation between norm and unit if “a versor is a quaternion of norm one (a unit quaternion)”?
a versor is a quaternion of norm one (a unit quaternion)
I want to understand relation between norm & unit (vector) because quaternion is a noncommutative division ring.
What is that 'unit' of ...
2
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1answer
44 views
Complement of a projective two sided ideal is two sided again?
If a finite dimensional algebra $A$ over a field $\mathbb{k}$ is semisimple then any two sided ideal of $A$ is generated (as a left module) by a central idempotent, so its (unique) complement is a two ...
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0answers
30 views
Fractional ideals of maximal orders in quaternion algebras
Let D be a skew field that is central and finite-dimensional over a number field F (in particular: a quaternion algebra over F). Let $\Delta$ $\subseteq$ D be a maximal $\mathcal{O}$$_{F}$-order. Let $...
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0answers
46 views
Counting Group of order n
I know how to count number of non isomorphic group in case of abelian case by Fundamental theorem of abelian case .I am interested in how to count non abelian non isomorphic group. I don't know is it ...
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0answers
20 views
Free module over $H$-module algebra
Let $H$ be a finite dimensional Hopf algebra, $R$ be a $H$-module algebra and $V$ be a finite dimensional $H$-module such that $R\otimes_{k} V$ is a finitely generated $R$ module under the action: $r.(...
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0answers
37 views
The radical and socle series of a module and its dual
I am study Peter Webb's book "A course in finite group representation theory", and stuck on ex.7 of chapter 6. The exercise is about the relationship between the socle series and radical series of a ...
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0answers
24 views
About left ideals of $M_2(D)$ [duplicate]
Can we characterize the left ideals of $M_2(D)$, the ring of $2\times 2$ matrices over a division ring $D$? How each non-trivial ideal of $M_2(D)$ is minimal?
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0answers
48 views
A characterization of left ideals in the ring of n×n matrices over a division ring
Please give me a solution for the following problem:
Let $D $ be a division ring and $n$ a positive integer. Then for each left ideal $I$ of $M_n (D) $, there are an invertible matrix $P$ and an ...
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0answers
29 views
Equivalent formulations of the Köthe’s Conjecture
Köthe’s Conjecture: If $Nil^*R =0$, then R has no nonzero nil one-sided ideals. ($Nil^*R$ is the sum of all nil ideals in R)
I am studying T.Y. Lam book and there is 2 equivalent formulations:
(A) ...
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2answers
244 views
$x^n = x$ implies commutativity, a universal algebraic proof?
I read in an answer on MO that Nathan Jacobson had given a universal algebraic proof that a ring satisfying the equation $x^n=x$ is commutative.
The sketch given in the answer is very clear : wlog ...
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1answer
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How to prove that $Nil_*M_n(R) = M_n(Nil_*R)$?
How to prove that $Nil_*M_n(R) = M_n(Nil_*R)$?,
$Nil_*R$ is the Baer's lower nilradical (It is the smallest semiprime ideal in R, and is equal to the intersection of all the prime ideals in R)
$M_n(...
2
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1answer
31 views
Loewy series of modules
I am studying the loewy series of module. But I can't understand how the series building.
I know that the socle is the sum of all simple submodules, but I can't understand the limit ordinal case.
2
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0answers
47 views
Differential operators and rules for Ore polynomials
When dealing with (nonlinear) dynamical systems, one often deals with state space representation, i.e. systems of the form
$$\dot{x}=f(x),\quad x(t)\in\mathbb{R}^n.$$
Let $x^*$ be a solution of this ...
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1answer
23 views
Any $R$-module is a quotient of some free module. Do we need “finitely generated” condition?
I'm reading Lang's Algebra on the section of semisimple ring. There is one proposition that says
If $R$ is semi simple, then every $R$-module is semi-simple.
Here we assume $R$ is unital but not ...
2
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1answer
36 views
Any matrix ring $R_k$ is isomorphic to a subring of the ring $R_{2^k}$
I am reading the paper: "Logical connections between some open problems concerning nil rings" by Jan Krempa.
I have a problem trying to understand the last part of Theorem 2. It says:
"Any matrix ring ...
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0answers
25 views
Tensor product of $K$-skew fields is nilpotent iff every $D_{i}$ is nilpotent?
an element, x, of a ring, R, is called nilpotent if there exists some positive integer, n, such that $x^n = 0$.
in book Skew Fields by P.K. DRAXL :
Definition 3 . A simple ring $A$ with finite $|A:Z(...
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0answers
28 views
A principal right ideal generated by an idempotent
One could decompose the right socle $S_r$ of a unital ring $R$ as $$S_r=S_1+S_2,$$ where $S_1$ is the sum of all nilpotent minimal right ideals of $R$ (with the nilpotency index 2) and $S_2$ is the ...
2
votes
0answers
38 views
Isomorphism between submodules as one-sided ideals
Let $R$ be a ring with 1, and $I$ and $J$ be left ideasl of $R$ such that they are isomorphic as left $R$-modules. Is it true that the left ldeals $I^2$ and $J^2$ are isomorphic ?
Thanks for any help!...
2
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0answers
49 views
About commutator operator of the Quaternion algebra
Let $\mathbb K$ be a field of $\text{Char}$($\mathbb K$)$\neq 2$.
Set $Q=Q(a,b\mid \mathbb K)=(a,b)_{\mathbb K}$ be the Quaternion algebra for $a,b \in \mathbb K$, with $\mathbb K$ basis $1, i, j$ and ...
1
vote
1answer
56 views
About isomorphism between quaternion algebras
Let $\mathbb K$ be a field of $\text{Char}$($\mathbb K$)$\neq 2$.
Set $Q=Q(a,b\mid \mathbb K)=(a,b)_{\mathbb K}$ be the Quaternion algebra for $a,b \in \mathbb K$, with $\mathbb K$ basis $1, i, j$ and ...
