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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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Relation between Symmetric algebra and Universal enveloping algebra as Lie algebras.

Let $L$ be a Lie algebra over $\mathbb{C}$. Assume $L$ satisfies PBW theorem. We can associate two Lie algebras with $L$: 1) $U(L):$ the universal enveloping algebra. Here the Lie bracket is defined ...
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Artin-Wedderburn theorem and representation theory

How to apply the artin-weddernburn therorem to representation theory? The artin-weddeburn theorem says that if $R$ is finite dimensional $k$ algebra( where $k$ is an algeraically closed field), then $...
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Is there any example of Artin algebra such that there Is a Gorenstein projective module that Is not strongly gorenstein projective

Is there any example of Artin algebra $A$ such that there exisits a Gorenstein projective $A$-module which Is not strongly Gorenstein projective $A$-module?
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Prime property in noncommutative rings without identity

Let $R$ be a ring (without assuming identity or commutativity), and $P$ a proper ideal of $R$. Show that the following are equivalent: (a) For ideals $A,B$: $AB\subseteq P$ implies $A\subseteq P$ ...
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Example III.3.2(1) of Baumslag's “Topics in Combinatorial Group Theory”: proving $F=\operatorname{gp}(1+\xi\mid \xi\in\Xi)$ is free.

This question is a little tricky (for me, at least), since in the textbook the proof of Theorem 5: Every subgroup of a free group is free. is not yet provided (even though I've seen such proofs ...
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multiplication of exponentials for non commuting matrices

Is there any special condition for the following statement to be true for any $n \times n$ matrices? $$ e^Ae^B=e^{A+B}=e^Be^A $$ Is it always correct to say that the exponential product equals ...
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Every finite dimensional algebra is subalgebra of a matrix algebra

I have two questions about the following exercise: Let $K$ be a field and $R$ a $K$-algebra. Assume that $d := dim_K R$ is finite. Show that there is an injective $K$-algebra homomorphism $...
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Prime Ideal In Left-Localization

Let $A$ be a left and right-Noetherian ring and let $S \subset A$ be left-localisable. Let $P \subset A$ be a prime ideal such that $P \cap S = \emptyset$. I wish to show $S^{-1}AP = \{\phi(s)^{-...
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Non-Commutative Algebra book

I am currently having a Master in topics such as Algebra, Geometry and Number Theory and recently I started studying Representation Theory where I've seen definition, theorems and propositions ...
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1answer
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Two Sided Ideal in Left-Localization of a Left-Noetherian Ring

Let $R$ be a left-Noetherian ring, and suppose $S \subset R$ is left-localizable ($S$ is assumed to be a multiplicative closed subset and $1 \in S$). Let $\phi: R \to S^{-1}R$ be the canonical ...
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Group commutator relations and the neutral element.

In Serge Lang: Algebra it is stated on page 69 that for a group $G$ that, if for $x,y,z\in G$ $y=[x,y]:=xyx^{-1}y^{-1}, z=[y,z], x=[z,x]$ is satisfied, we have $x=y=z=e$. I see that $y=[x,y]$ directly ...
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Why does this trick to derive the formula for $[A^n,B]$ in terms of repeated commutators work so well?

It is a known result that, given generically noncommuting operators $A,B$, we have $$ A^n B=\sum_{k=0}^n \binom{n}{k} \operatorname{ad}^k(A)(B) A^{n-k},\tag A $$ where $\operatorname{ad}^k(A)(B)\equiv[...
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Suggestion of research papers

I’m trying to find the research papers related to Hopf algebra over non commutative polynomial rings O( M_n(H)) to get concrete understanding about it. But unfortunately I couldn’t find any ...
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Matrix inverse algorithm that works for any unitary ring

Is there any algorithm to find out if a given square matrix has an inverse (which is both left and right inverse), and compute the inverse, if there is one for any unitary ring, without assuming ...
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associated graded ring is PI implies original ring is PI

This seems like it should be a known result but I didn't find it in a couple of standard references on noncommutative Noetherian rings. Recall that a unital associative ring $R$ is a polynomial ...
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Why is $\Omega (M)$ a superfluous submodule?

I'm struggling with a result that seems intuitive and that authors don't even bother to prove, so I think there's something stupid that I'm not seeing. Let $M$ be a finitely generated module over $...
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Localization of hereditary rings

A (left) hereditary ring $R$ is a ring for which all submodules of projective (left) $R$-modules are again projective. If $R$ is commutative, and $S\subseteq R$, then $S^{-1}R$ is hereditary :https://...
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1answer
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Facts about Ideals in Non-Commutative Rings

Let $R$ be a unital, but not necessarily commutative ring. Define its Jacobson radical $J$ to be the intersection of all maximal left ideals of $R$. Three questions: Let $R$ have unique two-sided ...
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1answer
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Example for quasi-commuting matrices

I can not find example of 2 matrices that quasi commute; $[A,B]=AB−BA = c I$, where $c$ is a scalar not equal to $0$ and $I$ is the identity matrix. As far as I know there is no 2x2 matrices that ...
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FP-injective , injective and pure R-module

prove that : FP-injective R-modules, which are pure-injective, are injective ? I know that an R-module M is called FP-injective if Ext1(N,M) = 0 for all finitely presented R-modules N and R-submodule ...
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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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1answer
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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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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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27 views

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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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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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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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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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
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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 ("...