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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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isoperimetric inequality hyperbolic groups

I am studying hyperbolic groups, and reading the proof of that: If a finitely presented group $\langle S|R\rangle$ satisfies a linear isoperimetric inequality, then it is hyperbolic. In the proof of "...
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Conceptual explanation for the identity of Hochschild about derivations

When reading Katz's paper "Algebraic Solutions of Differential Equations (p-Curvature and the Hodge Filtration)", he mentioned a mysterious identity about derivations in char $p$ commutative algebras ...
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Connection between properties of a ring and its quotient rings

Let $R$ be a $k$-algebra, $k$ a field, and $0\neq I \neq R$ a two-sided ideal of $R$. Denote by P a property of rings. One says that $R$ is 'just P', if $R$ does not satisfy property P but $R/I$ ...
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Proving that the Jacobson radical of a (not necessarily unital) ring is contained in the intersection of all left modular ideals.

I am following Lams book "A first course in non-commutative rings". I am attempting to prove that the Jacobson radical of a ring is precisely the intersection of all maximal modular left ideals of ...
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Surjective Ring Homomorphism mapping center to center

Let $\phi$ be an onto ring homomorphism that maps say $R$ to $S$. We know that under $\phi$, $Z(R)$ (the center of $R$, commutative elements under multiplication in $R$) get maps to a subset of $Z(S)$...
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Can you show that two matricies anticommute from the commutator?

I am trying to show that the pauli matricies anticommute, which is easy from direct commutation, but I cant do it from the commutation relation.
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Short exact sequence of bimodules

Let $A$ be a unital noncommutative algebra and let $$0 \longrightarrow M_1 \longrightarrow M_2 \longrightarrow M_3 \longrightarrow 0$$ be a short exact sequence of $A$-bimodules. I think if the ...
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A variation on Dixmier's counterexample concerning centralizers in $A_1$

I have asked this question in MO, but have not received any comments/answers, so now I ask it here: This question asks the following: "Suppose $k$ is a field of characteristic zero and $P$ and $Q$ ...
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Is there a calculus of non-commutative variables?

I know the integral calculus for commuting variables (which is just normal calculus). And there is an Grassman integral calculus for anti-commuting variables. (Variables where $\theta_1 \theta_2 = - ...
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Weyl algebra $\Bbb C[[x]][\partial]$ and division

Let $R = \Bbb C[[x]]$ the ring of formal power series and $A = R[\partial]$ the ring of differential operators with the relation $[\partial,x] = 1$. There is the following proposition in my book ("...
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Rings in which each element is a sum of $n$ commuting idempotents

Let $n$ be a nonnegative integer. Let $R$ be a nonunital ring such that every element of $R$ is a sum of $n$ pairwise commuting idempotents. (As usual, the class of nonunital rings includes the class ...
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Showing a certain endomorphism algebra is basic

Let $A$ be a finite dimensional algebra over $\mathbb{C}$. Let $P_1, \dots P_n$ be the projective indecomposable $A$-modules, pairwise non-isomorphic, and define $P=P_1\oplus\dots\oplus P_n$. Let $B=\...
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Why should a non-commutative operation even be called “multiplication”?

As per my knowledge and what was taught in school, $a\times b$ is $a$ times $b$ or $b$ times $a$ Obviously this is commutative as $a$ times $b$ and $b$ times $a$ are same thing. On the other ...
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Identity with repeatedly taking the commutator of a ring element

This is taken from Jacobson's Basic Algebra 2e, it's 2.1.5 If $a$ and $b$ are elements of a ring, define $a^{(0)} =a, a^{(1)} = [a,b] = ab-ba$ and $a^{(k)}=[a^{(k-1)},b]$ Prove the following formula: ...
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Pushout of unital non commutative algebras

I like to know if there is a pushout in the category of non commutative alegbras with unit and if the answer is "yes", who is it?
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coproduct of noncommutative algebra and commutative algebras

I have read the book "Rings with generalized identities" and I understand that the free product of asociative unital algebras are the coproduct of them, but I can't understand why this reduces to the ...
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How can I prove that for $2$ finite dimensional matrices $A,B: [A,B] = i$ is impossible?

I was told that for any finite dimensional Hilbert space, $[A,B]=i$ is impossible, while it is possible for an infinite dimensional Hilbert space. How can I show it is impossible for finite ...
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Wedderburn Artin theorem and module category

I am reading "An Introduction to Homological Algebra" by Rotman. In the chapter about adjoint functor, he says "The Wedderburn–Artin theorems can be better understood in the context of determining ...
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Showing that $(x+y)^2 \in A_1(k)$ does not have a Dixmier mate

Let $k$ be a field of characteristic zero, and denote the first Weyl algebra by $A_1(k)$, namely, $A_1(k)$ is the associative, non-commutative $k$-akgebra generated by $x$ and $y$ subject to the ...
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question about semisimple ring.

Wedderburn-Artin's theorem: A ring is left semisimple ring iff it is finite product of $M_{n_i}(D_i)$ for some division ring $D_i$. Due to this theorem,we know that if a ring is left semisimple,...
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V. Lebesgue counting points on hyperspheres - Use in Algebra?

[TL;DR] Any idea for an application in Non Commutative Algebra of V.Lebesgues formula for counting points on an hypersphere modulo p. ? Hi, I have recently encountered twice V.Lebesgues formula for ...
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On the essential ideal of a Lie algebra

everyone. I'm studying somethings about Lie algebras and I have a simple problem (apparently), but I don't get to prove. Question: How to prove that intersection of essential ideals of a semiprime ...
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$R$ semisimple $\implies M_n(R)$ is semsimple

I'm working out of TY Lam's first course in noncommutative rings. I'd like to show that if $R$ is a semismiple ring, $M_n(R)$ is as well. The obvious answer to me seems to be that since $R$ is a (left)...
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GRE 9768 #60 Boolean non-commutative rings: Prove $(-s)^2=s^2$ without commutativity.

GRE 9768 #60 Ian Coley's approach is to prove $(I)$ and $(I) \implies (II) \implies (III)$ In proving $(I)$, how do we prove $$(-s)^2=s^2$$ without commutativity (but with $s=s^2$, if need ...
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The definition of Jacobson radical

I see 2 definition of Jacobson radical in A First Course in Noncommutative Algebra of T.Y.Lam but I wonder if it is the same. Give $I$ is an ideal in $R$ called modular if there exist $e\in R$ such ...
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The left ideals of $R$ admit decompositions as sums left ideals whenever $R$ admits a decomposition as a sum of ideals [duplicate]

Let $R=B_1 \oplus \dots B_n$ where the $B_i$ are ideals of $R$. Then, there are idempotents $e_i \in B_i$ such that the $B_i$ are rings with identity $e_i$ and for $i \neq j$ $e_ie_j=0$. We wish to ...
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Rings isomorphic to a proper subring

Is there a theory for rings which are isomorphic to a proper subring? Which of the following rings have this property? $$ \mathbb{R} , M_2(\mathbb{R}) , \mathbb{C} \; and \; M_2(\mathbb{Z})$$
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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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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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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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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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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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$S_4$ 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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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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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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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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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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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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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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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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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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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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59 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 ...