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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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An example of a ring homomorphism

Before stating my question, let me recall some preliminaries in rings (especially noncommutative). Recall that for a noncommutative ring $R$, $‎\textbf{B}‎(R)=\{e\in Z(R): e^2=e\}$. $\textbf{...
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25 views

Deriving the projection onto the isotypic component

Let $k$ be a field, and $G$ a finite group such that the characteristic of $k$ does not divide $|G|$. Then $kG$ is a semisimple $k$-algebra, and the representation theory of $G$ over $k$ is semisimple....
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1answer
35 views

Proving that the first Weyl Algebra is simple

The first Weyl algebra over the complex numbers $\mathbb{C}[x]$ is defined to be the set $A_1 = {\{\sum_{i = 0}^{n} f_i(x) \delta^i : f_i(x) \in \mathbb{C}[x] }\}$. So it is the set of all linear ...
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1answer
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What are some simple examples of algebras? [closed]

So an algebra $A$ over a field $K$ is a ring under the operations $+$ and $x$ and also a vector space under a scalar operation and the operation $+$. What are some examples of algebras? When rings ...
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Why does “noncommutative probability” capture quantum probability?

In this article, Terry Tao states that non-commutative probability can be used for quantum probability. However, he then goes on to explain non-commutative probability without explaining how it ...
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1answer
58 views

Confusion regarding the definition of the Weyl Algebra

I've been studying algebraic structures recently; rings, fields, vector spaces and so on. I've recently just started learning what an algebra is, which, from what I can tell is a ring-like structure ...
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1answer
31 views

$S_4$ is not nilpotent but has central lower central series.

The lower central series of $S_4$ is given by : $$\gamma_1 =S_4\ge \gamma_2=A_4\ge\gamma_3=A_4\ge \gamma_4 =A_4\ge... $$ This series is clearly central as each $\gamma_i/\gamma_{i+1}$ is central in $...
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Definition of a $H$- stable ideal

Let $H$ be an Hopf algebra and let $A$ be an $H$-module algebra. What's the definition of a $H$-stable ideal?
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1answer
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The possible representations of a monoid derived from four “standard” or “regular” presentations and their relation

Given some monoid $M$ we can form the algebra $\mathbb C[M]$ by considering all formal sums and lineary extending the given multiplication in $M$. Let $f = \sum_{x\in M} \lambda_x x$ and $m\in M$. ...
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1answer
100 views

Matrices over noncommutative rings?

In chapter 1, section 2 of Categories for the Working Mathematician, Mac Lane says: For each commutative ring $K$, the set $\mathbf{Matr_K}$ of all rectangular matrices with entries in $K$ is a ...
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Non-constant coefficient matrix in first order linear differential equations

I want to solve a differential equation of the following form $$ \frac{d}{dt}x=A(t)x\, , $$ where $A(t)$ does not commute at different times. This equation holds on the interval $(a,b)$. Hence, the ...
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Classifying irreducible real representations

Let $G$ be a group and say $V$ is an irreducible representation over $\mathbb{R}$. Then $End_G(V) = End_{\mathbb{R}[G]}(V)$ must be a division algebra, since $V$ is a simple $\mathbb{R}[G]$-module. ...
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Direct sum of reals and quaternions is not a semigroup algebra for some semigroup

For a given semigroup $S$, the semigroup algebra over some field $F$ is the set of formal sum with the convolution product, and is denoted by $F[S]$. If we built the direct sum $$ \mathbb R \oplus \...
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32 views

Direct sum of Matrix algebras is not isomorphic to some semigroup algebra

An $n \times n$ matrix unit is any matrix which has zeros every, except at one position where it has one. By $E_{ij}^{(n)}$ we denote the $n \times n$ matrix unit which has its one at the $i$-th row ...
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1answer
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Every algebra could be decomposed into a part with unit, and without a unit. Question on uniqueness proof.

By an algebra $A$ over some field $F$ I mean a finite dimensional vector space over $F$ with an $F$-bilinear multiplication. That $A$ has a unit with respect to its multiplication is not assumed. A ...
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1answer
52 views

Complement of a maximal direct factor is indecomposable

Reading a book I saw the following assertion: Let $R$ be a ring (not necessarilly commutative) and $\varepsilon$ be the poset (w.r.t. inclusion) of all internal direct factors of an $R$-module $M$. ...
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2answers
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On the radical of a certain ideal of sixteen variable polynomial ring, generated by the entries of certain matrices

Consider the polynomial ring $R=\mathbb C[x_1,x_2,...,x_{16}]$, and set $$X=\begin{pmatrix} x_1 &x_2&x_3 &x_4\\ x_5&x_6& x_7&x_8\\x_9&x_{10}&x_{11}&x_{12}\\x_{13}&...
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1answer
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Question on proof that matrix algebra over given algebra is semisimple iff original algebra is semisimple

Let $A$ be a finite-dimensional linear associative algebra over some field $F$. Then denote by $M_n(A)$ the set of $n \times n$ matrices with entries in $A$ and the usual operations. Then $M_n(A)$ ...
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66 views

Find the Jacobson radical of a matrix ring

Find the Jacobson radical of matrix ring $R=\mathbf{M}_2(K\otimes_kK)$ where $k =\mathbb{F}_2[x]$ and $K = \mathbb{F}_2[x^{1/2}]$ I tried to find the radical of this matrix ring. I have also found ...
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A question about semiregularity

Let $R$ be a unital ring such that any cyclic right ideal is the direct sum of a ring direct summand $eR$ and a right ideal $S$ of $R$ with $S\subseteq J(R)$, where $e=e^2$ and $ J(R)$ is Jacobson ...
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2answers
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Is the $\mathbb{Z}$-grading of a Clifford algebra basis independent?

Let $V$ be a finite dimensional vector space over a field $K$ of characteristic $\neq 2$, and let $q \colon V \to K$ be a quadratic form. One of the first things to show when learning the theory of ...
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Associative, non-commutative, non-trivial, analytic binary operation

There was a question whether associative, but non-commutative binary operation over the real numbers exist. A trivial answer is the binary operation $x\circ y = x$ or $x\circ y = y$. As a followup, ...
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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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1answer
57 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
28 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
49 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
96 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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32 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
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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
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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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2answers
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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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1answer
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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
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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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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
26 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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1answer
70 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
37 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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48 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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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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1answer
268 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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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
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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
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2answers
72 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
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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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29 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 $\...