Questions tagged [noncommutative-algebra]
For questions about rings which are not necessarily commutative and modules over such rings.
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How to use BCH formula to compute mutiplication of group element?
i cant get the right answer.
Given the commute rule of (for example, the caculation below will be in static group):
we have: generator H, P, K, J, and commutation rule $[H,P]=0$, $[H,K]=0$, $[P,P]=0$...
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Finiteness of results in Connes-Kreimer approach
Remark: Although this is technically a physics-related post, the content heavily relies on pure mathematics, so I deemed it more appropriate here.
When reading the papers by Connes and Kreimer (e.g. [...
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Nonunital non commutative ring with 3 ideals...
It is well known that if a (unital commutative) ring A has only three ideals ({0}, J, A), then the quotient A/J is a field.
But, what can we conclude about A/J if A is not commutative nor unital but ...
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Polynomial of a ring is nilpotent if all of its coefficients are nilpotent? [duplicate]
Let $R$ be a ring (not necessarily commutative) and $R[x]$ its polynomial ring. Suppose $f(x)\in R[x]$ is such that $f(x)=a_0+a_1x+\cdot\cdot\cdot + a_nx^n$. My question is if $a_i$ is nilpotent for ...
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Finite dimensional Irreps (of algebras) with same traces must be equivalent ('page 136' in Bourbaki)
I look for the reference (or proof) of the following fact which is from appendix (B $27$) of Dixmier's book on $C^*$-algebras.
Claim: Let $A$ be an algebra (not necessarily commutative) over a field $...
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Tensor over matrices [duplicate]
Suppose $A$ and $B$ be two ring. Show that $M_n(A) \otimes_{\mathbb{Z}} M_m(B) \simeq M_{nm}(A \otimes_{\mathbb{Z}} B)$.
Actually if define $\psi : M_n(A) \times M_m(B) \rightarrow M_{nm}(A \otimes B)$...
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Conceptual definition of the Auslander-Reiten translate
In homological algebra, we learn to differentiate between
The conceptual definition.
A computation, which is done by choosing efficient resolutions.
The only definition I've seen of the Auslander-...
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Noncommutative analogue of ring of integers in number field
I'm studying Igor V. Nikolaev's paper "Untying knots in 4D and Wedderburn’s theorem". In the paper, he works with hyper-algebraic fields $\mathbb{K}$, i.e., fields with noncommutative ...
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Are primitive ideals left maximal in a non-commutative ring?
Let $\mathcal{R}$ ring and $I \subset \mathcal{R} $. $I$ is called a primitive ideal of $\mathcal{R}$ if it is a left ideal and $\exists \, M$ simple left $\mathcal{R}- $module such that $Ann(M)=I$. ...
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Why do Kontsevich and Rosenberg use algebra epimorphisms rather than surjections?
In the article Noncommutative spaces Kontsevich and Rosenberg make the following definition:
2.6. The Q-category of infinitesimal algebra epimorphisms. Let $A$ be the
category $Alg_k$ of associative ...
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Ideal generated by central element is central in a prime ring?
A ring $R$ is said to be prime if $xRy=0 \implies x=0$ or $y=0.$
Let $R$ be a prime ring and $x \in Z(R)$. Then the ideal generated by $x$ is central i. e. $\langle x \rangle \subseteq Z(R).$
I ...
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Wedderburn Artin's theorem for algebras over a field
I learned about the Wedderburn Artin's theorem for simple left artinian ring, says that if $R$ is simple left Artinian ring then $R\cong\mathrm{M}_n(\Delta)$, for some division ring $\Delta$.
I want ...
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If $R$ is Armendariz ring then its prime radical $P(R)$ is Armendariz or not?
First let us write about the definitions of some terms.
Armendariz ring : A ring $R$ is said to be Armendariz if whenever $f(x),g(x)\in R[x]$ satisfy $f(x)g(x)=0$ then $a_ib_j=0$ for all $i,j$. Here $...
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On central simple algebras and the Wedderburn Artin's theorem
I learned about the Wedderburn Artin's theorem, says that if $R$ is simple left Artinian ring then $R\cong\mathrm{M}_n(\Delta)$, for some division ring $\Delta$.
Now, I was studying the Brauer group. ...
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About $(\frac{a,1-a}{k})\cong\mathrm{M}_2(k).$
$k$ is a field.
A quaternion algebra over $k$ is a $4$-dimensional $k$-algebra with a basis $1,i,j,ij$ with the following multiplicative relations: $i^2\in k^\times, j\in k^\times, ij=-ji$ and every $...
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