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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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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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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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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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 ...
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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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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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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 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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Commutation relation in Algebraic Bethe Ansatz

In the Algebraic Bethe Ansatz paper https://arxiv.org/abs/hep-th/9605187, when I try to take the limit $\ \lim \mu \to \infty$ to derive the equation (98), I get the commutation relation as $\sum_{\...
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Can you generalise the Chinese Remainder Theorem to noncommutative rings without identity?

Ultimately, my question is: does the following theorem hold? Let $I_1, ..., I_n$ be ideals of some ring $R$, with $R = I_i + I_j$ for $1 \leq i < j \leq n$. Then for any $r_1, ..., r_n \in R$ ...
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Matrices satisfying some conditions

I'm looking for examples of matrices $u$, $v$, and $Q$ satisfying the following conditions: $u$, $v$, and $Q$ are relatively small matrices of the same size (perhaps 2 by 2, 3 by 3, or 4 by 4) $u$ ...
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Free product of quotients of free algebras

Let $k$ be a commutative ring and let $A=k\langle X_1,...,X_n\rangle/(f_i\mid i\in I)$, $B=k\langle Y_1,...,Y_m\rangle/(g_j\mid j\in J)$ be quotients of free noncommutative $k$-algebras. Question: Is ...
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$soc^n(M)=\{m\in M | mJ_A^n=\{0\}\}$?

Consider a right $A$-module $M$ over an Artinian ring $A$ (unital, not necessarily commutative). Define $soc(M)$ to be sum of all simple $A$-submodules of $M$. Inductively define $soc^0(M):=0$, $soc^n(...
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A graded Baker-Campbell-Hausdorff formula

Let $A$ be a graded algebra. Let $a,b\in A$ such that $\deg(a)=1$ and $\deg(b)=0$. I am looking for a "nice" expression for the degree 1 part of $e^{a+b}$. In other words, let $$e^{a+b}=\...
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2 answers
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Ring with three idempotents satisfying $\alpha+ \beta+ \gamma=0$. Prove that $\alpha=\beta= \gamma$.

Problem Let be $(R,+,\cdot)$ a ring. Moreover, $(R,+,\cdot)$ has the property that if $$2022x=0$$ then $$x=0.$$ Let be $\alpha$, $\beta$, $\gamma$ three idempotent elements such that $$\alpha+ \beta+ ...
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  • 1,025
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 ...
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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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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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1 answer
76 views

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
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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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  • 353
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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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
0 answers
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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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  • 1,148
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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 ...
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1 vote
0 answers
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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 ...
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1 vote
0 answers
31 views

Commutative subalgebras of a matrix algebra over a division algebra

Let $K$ be a field of characteristic $0$, $D$ a central semi-simple division algebra of dimension $d^2$ over $K$, and $n$ a positive integer. Let $R$ be a maximal subfield of $M_n(D)$, then is $\dim_K ...
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  • 279
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0 answers
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If $\ker(f_i)\cap f_i(M)=0$ for each $i=1,\cdots, n$, then is it true that ker$(I)\cap I(M)=0$ for any f.g left ideal $I$ of $\text{End}_R(M)$?

Let $R$ be a ring with identity, $M$ a right $R$-module and $S=\text{End}_R(M)$ the ring of $R$-endomorphisms of $M$. Let $I\leq{} _{S}S$ be any nonzero left ideal with a finite number of generators ...
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2 votes
1 answer
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Do $D$-modules necessarily have the "composition" condition?

This might be a stupid question. Every bone in my body tells be that if I have a ring $R$ and a right $R$-module $M$ then we should have $(m\cdot r)\cdot s= m\cdot(rs).$ However, I have just done this ...
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  • 2,896
1 vote
0 answers
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"Algebraic dimension" for finite-dimensional (non-associative) algebras?

Let $V$ be some finite-dimensional vector space (over some field $\mathbb{K}$), then a (possibly non-associative) algebra $A$ on $V$ corresponds to a bilinear map $V \times V \to V$. I prefer answers ...
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1 vote
1 answer
56 views

The group ring of the dihedral group $D_6$

I'll use the $D_6=\langle f,r\mid f^2,r^3,frfr\rangle$ definition for $D_6$. I'm trying to study the structure of $\mathbb Z[D_6]$. Units One thing I've noticed $(1+fr)^2=1+2fr+(fr)^2=2(1+fr)$, ...
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2 votes
0 answers
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Is centralizing all idempotents just once enough to make a ring abelian?

A ring $R$ is called abelian if all idempotents in $R$ are central. So, every ring should have a universal abelian quotient ring. One way to do this is to construct an ascending chain of ideals $I_0 \...
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9 votes
0 answers
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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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Evaluating the sum over all strings made of two anticommuting terms

Given two anticommuting elements, $A$ and $B$, I aim at evaluating the sum over all strings of length $n$ multiplying exactly $k$ elements $A$ and $n-k$ elements $B$ (as we know, there are $\binom{n}{...
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1 vote
0 answers
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Expanding $(Ae+Bf+Ch)^n$ for $\mathfrak{sl}_2$ generators $e,f,h$

I have to deal with an expression of the form, $$(Ae+Bf+Ch)^n$$ where $n \geq 0$ and $A,B,C \in \mathbb C$ where $e,f,h$ is the usual Chevalley basis of $\mathfrak{sl}_2$, i.e. $$[e,f] = h, \quad [h,e]...
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  • 1,641
1 vote
1 answer
37 views

How is it possible to rewrite the matrix product as a commutator in a Lie algebra?

so suposing $(t^m)_{m \in M}$ are the generators of a Lie group, and we have the following relation in the Lie algebra: $$[t^a,t^b]= i f^{abc}t^c$$ Where the $f^{abc}$ are the structure constants of ...
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5 votes
1 answer
148 views

Relation between the dimensions of the fixed point and the submodule given by the augmentation ideal

Let $k$ be a field with characteristic $p>0$, let $G$ be a finite $p$-group of order $n=p^m$, and let $W$ be a (left) $k[G]$-module such that the $\dim_k(W)=n$. If we denote by $I$ the augmentation ...
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  • 1,130
1 vote
0 answers
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$T(z) e^{-\partial_z} $ for Yangian is a Manin matrix

Let $T(u)$ be the generating matrix of the Yangian $Y(\mathfrak{gl}_n)$ of $\mathfrak{gl}_n$. So we have the identity $[T_{ij}(u),T_{kl}(v)]=\frac{1}{u-v}(T_{kj}(u)T_{il}(v)-T_{kj}(v)T_{il}(u))$. We ...
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  • 105
1 vote
0 answers
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exponential of non-commuting matrices

In general, let $A$ be a Banach Algebra, we have that $\exp(a+b) = \exp(a) + \exp(b)$ if $a$ and $b$ commute. I'm interested in the case of non-commuting $a$ and $b$ at the neighborhood of $0_A$. ...
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  • 710
0 votes
0 answers
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Exponential of a sum in a non-commutative graded algebra

Let $a,b$ be two elements of a graded algebra $A$ such that $\deg(a)=1$, $\deg(b)=0$ and $[a,b]\neq 0$. I would like to know whether there exits an explicit expression for the degree 1 component $$\...
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  • 857
4 votes
1 answer
36 views

Is $\mathcal{D}_k(k[x])$ generated by $\partial_1,\partial_p,\partial_{p^2},\ldots$ and multiplication maps?

Let $k$ be a field of characteristic $p>0$, and let $\mathcal{D}=\mathcal{D}_k(k[x])$ be the ring of differential operators on $k[x]$. That is, we define $$ \mathcal{D}_{\leq -1}:=\{0\}\subseteq\...
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1 vote
1 answer
64 views

Products of Non-Commuting (or orthogonal) Projections

Let $S=\{P_1,P_2,\dots,P_n\}$ be a set of Hilbert space projections $P_i=P_i^2=P_i^*$ that only commute when they are orthogonal or equal: $$P_iP_j=P_jP_i\implies P_iP_j=0\text{ or }i=j.$$ Let $i,j,k$ ...
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  • 25
2 votes
1 answer
142 views

Existence of a finite extension of the primes

Let $R$ be a ring (possibly non-commutative with zero-divisors). A non-unit and non-zero-divisor element $r \in R$ will be called irreducible if for all $a,b \in R$ such that $r=ab$, then $a$ or $b$ ...
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1 vote
1 answer
49 views

Commuting polynomials in twisted polynomial ring with constant terms satisfying a polynomial relation

Suppose $q$ is a prime power, and let $A=\mathbb{F}_q[x,y]/f(x,y)$ where $f(x,y)$ is an irreducible polynomial. Let $K$ be any field such that $A$ injectively maps into $K$. (For ease of notation, ...
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0 votes
2 answers
40 views

Why is addition not commutative under PM's notion of relation number?

Quoting Bertrand Russell's "The Principles of Mathematics" p468 §299: It is worth while to repeat the definitions of general notions involved in terms of what may be called relation-...
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0 answers
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How to show that $x^n\notin (xy-yx-1)$ inside $k\langle x,y\rangle$?

This may be a trivial question, but somehow I just can't seem to find an easy enough argument, and I lose myself in complicated constructions. Let $k$ be a field and let $k\langle x,y\rangle$ be the ...
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0 votes
1 answer
95 views

When is Tor(M) a submodule?

I am reading about torsion of a module over a commutative ring. Definition: $\text{Tor(M)} = \{ x \in M | rx = 0 \text{for some non zero r} \in R $ where $R$ is an integral domain. This is a submodule....
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