# Questions tagged [noncommutative-algebra]

For questions about rings which are not necessarily commutative and modules over such rings.

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### Two sided ideals of $k\left<x, y\right>$

Question: Are the two sided ideals of $k\left<x,y\right>$ (polynomial ring in twonon commuting variables) finitely generated (as two sided ideals) when $k$ is a field? I know that there are one ...
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2 votes
2 answers
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### Every finite-dimensional algebra which is not simple contains a maximal ideal whose annihilator is nonzero

The following problem is from Chapter 3 of Drozd and Kirichenko's "Finite-Dimensional Algberas" that I am self-studying. Let $A$ be a finite-dimensional unital algebra that is not simple. ...
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### Isomorphism in the quotient ring of a localization

Let $R$ be a ring with unity and suppose $I$ is a maximal ideal of $R$ such that $M = R \setminus I$ is a right denominator set for $R$. Is it true that $R_M/I_M \simeq R/I$, where $R_M$ is the right ...
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0 votes
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### Dimension of central divinsion algebra

Let A be a finite dimensional central division F-algebra such that [x, y][z,w]+ [z,w][x, y] $\,\in \,$F for all x, y, z,w $\in$ A. Prove that either A = F or [A : F] = 4. Here, [x, y]=xy-yx, the ...
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7 votes
1 answer
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### Find $A^m B^n$ for noncommutative variables

Let $$\begin{cases} AC=CA+\alpha A,\\ BC=CB+\beta B,\\ AB=BA+\gamma C. \end{cases}$$ It is no so hard to find that \begin{gather*} A^n C^m=(C+\alpha n)^m A^n,\\ B^n C^m=(C+\beta n)^m B^n. \end{...
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1 vote
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### Do matrices over noncommutative division rings have well-defined ranks?

It is known that the row and column rank of any matrix over a field are the same and their common value is simply called the rank of the matrix. Now, for any $m$-by-$n$ matrix $A$ with entries in a ...
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1 vote
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### The module of section of a vector bundle and the module of section of the pullbackbundle are isomorphic

Let $\pi: E \rightarrow M$ be a vector bundle and let $f: N \rightarrow M$ be a continuous map. Let $f^{*} E$ be pullback bundle Let $\Gamma(E)$ be the module of section of the vector bundle $E$ and ...
0 votes
0 answers
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### Formula for matrix inverse with non-commutative entries [duplicate]

I have a square matrix $A$ with elements $A_{i,j}\in\mathbb{A}$ where $\mathbb{A}$ is a ring with with addition ($+$) and multiplication operations ($\times$). The operation $\times$ is non-...
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1 vote
1 answer
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### Left ideals of group algebra $K[\mathbb H]$

I've been studying Hopf-Galois theory, and while writing by myself certain example, I came up with this question: Let $K$ be a field, and $\mathbb H$ the quaternion group. Is $K[\mathbb H]$ the only ...
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1 vote
1 answer
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### Absolutely irreducible/simple $A$-module iff Endomorphism ring consists of scalar matrices

Let $A$ be a non-commutative $K$-algebra (where $K$ a field), whose underlying $K$-vector space is finite dimensional. Definition An $A$-module $M$ is said to be absolutely irreducible or abs. simple ...
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1 vote
1 answer
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### Clifford algebra of a free bimodule over a noncommutative ring

I have an apparent definition (and construction) of the Clifford algebra of a free $R$-$R$-bimodule $M$ with a quadratic form $q: M \rightarrow R$ with noncommutative $R$. I am not aware of any ...
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1 vote
1 answer
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### Can a matrix ring have a non-free ring extension?

Let $D$ be a skew field, and consider an arbitrary noncommutative unital ring $R$ extending the matrix ring $D^{n\times n}$. Must $R$ be free as a $D^{n\times n}$-module? If $R$ is finite-dimensional,...
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### Nontrivial converse to Skolem-Noether theorem?

The Skolem-Noether theorem says that if $A$ is a k-central simple k-algebra, then $Aut(A) \cong Inn(A)$. Is there a $k$-algebra $A$ such that $$Aut(A) \cong Inn(A) \not \cong \{*\}$$ but $A$ is not a ...
5 votes
0 answers
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### Freyd-Mitchell embedding theorem with commutative rings

For a small abelian category ${\cal A}$, the Freyd-Mitchell theorem guarantees that ${\cal A}$ is equivalent to a full subcategory of ${\bf Mod}_R$ for some ring $R$ in a way that preserves exactness. ...
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0 votes
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### Metric connection as map from $\Omega^1 \, \to \, \Omega^1 \otimes \, \Omega^1$

I was reading Shahn Majid's and Edwin Begg's book on Quantum Reimannian Geometry. In this they define the metric to be a map $$g:\Omega^1 \otimes \Omega^1 \to A.$$ I do not understand this definition. ...
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0 votes
1 answer
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### An $R$-division ring $(\mathcal{D},\varphi)$ is epic iff the division closure of $\varphi(R)$ in $\mathcal{D}$ is $\mathcal{D}$.

I am studying non commuatitve ring theory and I'm having problems with the following exercise: Let $R$ be a ring and $(\mathcal{D},\varphi)$ an $R$-division ring. Then, $(\mathcal{D},\varphi)$ is epic ...
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5 votes
1 answer
45 views

### Questions regarding rings where every injective modules over them are also flat modules.

Let $R$ be a ring such that every left injective $M$ module over $R$ is also a left flat module over $R$. An example that springs to my mind to illustrate this type of rings are the regular rings, ...
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4 votes
0 answers
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### Formal derivative, any problems for non-commutative rings?

Let $R$ be a ring and $R\left[ x \right]$ be the ring of polynomials over $R$. If $f=a_nx^n+\cdots+a_0 \in R\left[ x \right]$, we define the formal derivative $f^{'}=na_nx^{n-1}+\cdots+a_1$. Let $f$ , ...
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0 votes
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### Global dimension of a quiver with relations that do not overlap

I am reading Green, Hille & Schroll's paper Algebras and Varieties and find myself a bit confused with Proposition 5.1, which says: Consider a field $K$, a quiver $Q$ and a tip-reduced nonempty ...
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3 votes
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### Is it true that $M_n(R)\otimes_RS\cong M_n(S)$ for noncommutative rings?

It is easy to see that it is true for commutative rings under the canonical map $A\otimes s\mapsto sA$. When I wanted to generalize to noncommutative ring. Then I see this map is not really ...
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0 votes
1 answer
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### Separating in even and odd powers if they don't commute

During a lesson on Rabi's oscillations, my professor computed the following series: $$H = \sum_n \left( (a |e_1 \rangle \langle e_0| + a^\dagger |e_0 \rangle \langle e_1|\right)^n$$ separating in odd ...
1 vote
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### Exponential/binomial theorem of sum of non-commuting operators

I'm looking at some non-commuting operators $A, B$ say and I'm interested in their exponential $e^{A + B}$ or formulas for binomial expansions like $(A + B)^n$ when the operators have a nice ...
1 vote
0 answers
31 views

• 7,965
9 votes
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### Composition of some linear differential operators $(D-A_n)...(D-A_1)$

Let $D=x\frac{d}{dx}$ and $A_i\in\mathbb{R}[[x]]$ for $i=1,...,n$. Let $B_i\in\mathbb{R}[[x]]$ for $i=1,...,n$ such that $$(D-A_n)...(D-A_1)=D^n+\sum_{i=1}^nB_iD^{n-i}.$$ Is there a well-known formula ...
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1 vote
1 answer
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