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Questions tagged [noncommutative-algebra]

For questions about rings which are not necessarily commutative and modules over such rings.

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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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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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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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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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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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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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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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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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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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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 $\mathbb{k}$-...
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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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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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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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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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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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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 ...
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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 ...
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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 ...
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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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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
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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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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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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 ...
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1answer
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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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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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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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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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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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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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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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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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$x^n = x$ 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(...
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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.
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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 ...
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1answer
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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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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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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 ...
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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!...
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0answers
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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 ...
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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 ...