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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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Existence of conjugation on a quaternion algebra given a separable subalgebra

In Vignéras' book Arithmétique des algèbres de quaternions, a quaternion algebra $A$ over a field $K$ is defined as a $4$-dim central algebra for which there exists a separable $2$-dim (necessarily ...
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What properties of $R$ does the monoid ring $R[M]$ inherit?

It's known that polynomial rings $R[x_1,...,x_n]$ inherit some properties of a base ring $R$. For example, (due to the wikipedia article on polynomial rings) if $R$ is an integral domain, then so is $...
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When $\bigcap_{i=1}^n I_i =\prod_{i=1}^n I_i$ for any (noncommutative) ring?

In some thesis there are given ideals $I_i \subset R$ which are pair comaximal and generated by central elements of ring $R$ and it's written "then $\bigcap_{i=1}^n I_i =\prod_{i=1}^n I_i$ for any $n\...
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Concerning the Dixmier and the Jacobian Conjectures

I have also asked my following question in MO: Denote by $W$ the first Weyl algebra over a field $K$ of characteristic zero, $W := \langle X,Y | YX-XY=1 \rangle$. Based on Guccione, Guccione, and ...
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Exponentiation in Non-Commutative Rings

I'm fairly new to abstract algebra and in an exercise I was asked under which conditions it is true that $(ab)^n = a^nb^n$, for $a,b \in R$ and $n$ a positive integer, where $R$ is a ring. It can be ...
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Question regarding flat localization of left modules

This is taken from Rosenberg's non-commutative affine scheme https://sasharosenberg.com/?x-portfolio=noncommutative-affine-schemes page 3-4, where he talks about flat localizations of $R$-$Mod$. Let ...
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Show that this submodule is not completely reducible

Let $M = \begin{bmatrix} \mathbb{C} \\ \mathbb{C} \end{bmatrix}$. Show that $M$ is not completely reducible as a left module over $R = \begin{bmatrix} \mathbb{C} & 0 \\ \mathbb{C} & \mathbb{C}...
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Group rings decomposition

Let $G$ be a group with $|G| = 8$. By the Artin-Wedderburn Theorem, $\mathbb{C}G$ is isomorphic to the direct sum of matrix rings over division rings. What are the possible choices for a ...
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Powers of ideals in group ring $\mathbb{Z}_nS_m$

Let the group ring $\mathbb{Z}_2S_3,$ where $S_3$ is the permutation group on $3$ elements, with presentation $S_3 = \{1, \sigma, \sigma^2, \tau, \sigma\tau, \sigma^2\tau \} = \langle \sigma, \tau | \...
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Is there math with non-commutative multiplication of real numbers?

I'm wondering is there a math with non-commutative multiplication of real numbers. For example, we could define operator ⊗ for $ n, m ≥ 0$: $$ n⊗ m = n\times m $$ $$ n⊗ (-m) = n\times m $$ $$ -n⊗ m ...
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suggest a course on differential operator and Weyl

Can someone suggest me a course in pdf or textbook about differential operator and Weyl algebra. let $ R$ a commutative algebra over a field $k$ let M, N two R-modules. we define the set of ...
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1answer
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For an Artinian ring semiprimitive implies semisimple.

I'm currently reading Rotman's An Introduction to Homological Algebra (2nd edition), and on page 188 in the proof of Theorem 4.66 (Every left Artinian ring is semiperfect), I ran across the claim: ...
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1answer
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Under what assumptions on the grading is the fundamental theorem of twisting morphisms true?

In my favorite book Algebraic Operads by Loday and Vallette there is a theorem (2.3.1 in my book) which says that for a twisting morphism $\alpha:C\to A$ from a connected wdga coalgebra to a connected ...
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1answer
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System of homogen linear equations in a division ring

Let $K$ be a division ring (one does not suppose that $K$ is commutative) and $m,n$ two positive integers such that $m<n$. Consider the system of homogen linear equations $$\left\{\begin{array}{rcl}...
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Notion of 2-sided artinian ring?

An Artinian ring is defined to be a ring that is both left Artinian and right Artinian. So satisfies Descending chain condition (DCC) on left and right ideals. This implies it also satisfies DCC for ...
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non commuative division ring with a discrete valuation

I do not manage to exhib a non commutative ring of division with a discrete valuation. Can anyone show me one? Examples with quaternions would be a plus !! Thanks in advance.
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Is $\mathbb Z_3+\mathbb Z_3\mathbf i+\mathbb Z_3\mathbf j+\mathbb Z_3\mathbf k$ a local ring?

I consider the quaternion division ring on $\mathbb Q_3$: that is $$\mathbb H_{\mathbb Q_3}=\{a+b\mathbf i+c\mathbf j+d\mathbf k \mid a,b,c,d\in\mathbb Q_3\}$$ with $\mathbf i^2=\mathbf j^2=\mathbf k^...
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The sum of two non-units of a ring $R$ is a non-unit implies that the Jacobson radical is maximal.

My problem is: If the sum of to non-units in a ring $R$ is non-unit, then the Jacobson radical $J(R)$ is maximal. I need help please. I have no idea how to start. I thought that if the set of all non-...
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If $(e^{\alpha(t)})'=\alpha'(t)e^{\alpha(t)}$ then $\alpha(t), \alpha'(t)$ commute?

Let $\alpha(t)$ be a smooth path of real $n \times n$ matrices. (Formally $\alpha:(-\epsilon,\epsilon) \to M_n(\mathbb{R})$). If $(e^{\alpha(t)})'=\alpha'(t)e^{\alpha(t)}$ for every $t$ or $(e^{\...
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3answers
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If $ab - ba = c, bc - cb = a, ca - ac = b$ in a ring, prove that $a^2 + b^2 + c^2$ commutes with $a, b$ and $c$.

I am stuck with a problem about a non-commutative ring. (I am rather new with abstract algebra.) By only putting $a, b, c$ in their expanded forms into equation $a (a^2 + b^2 + c^2) = (a^2 + b^2 + c^...
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1answer
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Representation of an algebra is absolutely irreducible if and only the representation map is surjective

This should be well known but I can't seem to locate a reference: Let $k$ be a field, $V$ a $n$-dimensional vector space over $k$ with an action of a $k$-algebra $A$. We say that $V$ is an absolutely ...
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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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1answer
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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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1answer
68 views

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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1answer
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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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1answer
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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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1answer
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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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1answer
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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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1answer
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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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1answer
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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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1answer
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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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2answers
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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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0answers
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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....