Questions tagged [noncommutative-algebra]

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

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0answers
17 views

Non-commutative Archimedean-ordered ring without an unit [closed]

Give an example of a non-commutative Archimedean-ordered ring without an unit.
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19 views

Direct sum of simple modules is cyclic [duplicate]

Let $R$ a ring, and $M$, $N$ simple $R$-modules non isomorph. Prove that $M\oplus N$ is cyclic. I am confused. If $(m,n)\neq 0$, $\phi: r\mapsto (m,n)r$ and the projections $\pi_i$ are R-homomorphism. ...
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48 views

Do Artin rings contain many commutative Artin subrings?

Suppose $R$ is a Artin ring and $x\in R$. Is there necessarily a commutative subring of $R$ which is also Artin and contains $x$? What if I want it to contain finitely many commuting elements $x_1,\...
2
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1answer
51 views

Loewy decomposition of differential operators

The paper by Fritz Schwarz, "Loewy decomposition of linear differential equations", contains the following lemma, which I try to prove in order to understand the algorithm which Schwarz ...
5
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0answers
98 views

Why does the exactness of a Koszul complex require commutativity?

Most references on Koszul complexes seem to assume that the elements $x_1,\ldots, x_n$ live in a commutative ring or are central. It appears to me that the proof also works provided the weaker ...
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1answer
33 views

What is the smallest dimension a non-commutative C*-algebra can have?

What is the smallest dimension a non-commutative C-star-algebra can have? Let $d$ denote this dimension. Clearly, $d\leq 4$ as $M_{2}(\mathbb{C})$ is a $4$-dimensional non-commutative C-star-algebra. ...
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18 views

Obtaining the product of any element with a Hamiltonian path in a Cayley graph [closed]

Consider the Cayley graph of a finite non abelian group, generated by a generating set, say $\{s,t\}$. Let me explain my question by thinking of a small example. Let the Cayley graph have 5 vertices, $...
5
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2answers
121 views

Matrix exponential for non-commutative operator entries of matrix

I would like to find the matrix exponential $e^{iHt}$ of the Hermitian matrix $H$ where $$ H=\begin{pmatrix} \delta& \sqrt{2}a & 0\\ \sqrt{2}a^\dagger &0& \sqrt{2}a\\ 0 &\sqrt{2}a^\...
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25 views

Finitistic dimension conjeture for $A^{op} $ implies the strong Nakayama conjecture for A

I have some trouble with some detail in the proof of the following theorem. Assume that the Finitistic dimension conjecture is true for $ A^{op} $ that is $ sup\{ proj.dim(M) \vert M \in mod(A^{op}) ~...
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94 views

Non-central tensor product of central algebras

If $K$ is a field, it is easy to show using a basis that if $A$ and $B$ are $K$-algebras, then $Z(A\otimes_K B) = Z(A)\otimes_K Z(B)$ (where $Z$ denotes the center). This is no longer true if we ...
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1answer
33 views

Do the matrix ring endofunctors have left adjoints?

Given any positive integer $n$, does the endofunctor $R \mapsto M_n(R)$ of the category of (unital) rings have a left adjoint? One possible idea for constructing the left adjoint $L(R)$ at a ring $R$ ...
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26 views

Does Hochschild (co)homology preserve quasi-isomorphisms?

Does Hochschild (co)homology preserve quasi-isomorphisms? I.e. if we have an algebra in chain complexes $A$ and a chain complex $M$ that is a bimodule over $A$, we may form the cyclic simplicial ...
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1answer
57 views

Can we define “Algebra of fractions” like ring/field of fractions?

According to the answers to this question, if $R$ is a ring with no zero divisors, we can define a ring of fractions of $R$ if $R$ satisfies the Ore condition and that ring will be a skew-field. What ...
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1answer
73 views

Baker–Campbell–Hausdorff formula for generators of $SO(3)$

By noting ${e_1 = (1, 0, 0), e_2 = (0, 1, 0), e_3 = (0, 0, 1)}$ and $E_i = [e_i]_\times$ the generators of $SO(3)$, then we have the following commutator properties : $$ \begin{align} E_1 &= [...
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1answer
30 views

The $q$ Multilinear Theorem

Let $R$ be the skew polynomial ring $k_\mathfrak{q}[x_1,\ldots,x_m]$ where $x_ix_j=qx_jx_i$ with $q\in k^*$ and for all $i<j$. The $q$ Multinomial Theorem states that $$(x_1+\ldots+x_m)^r=\Sigma_{...
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84 views

Common multiples in a group ring

Consider the group ring $\mathbb{Q} F_2$ with rational coefficients over the free group in two generators ($a$ and $b$). It is not an Ore-Domain: While it does not contain any zero-divisors, for ...
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31 views

Quaternion Algebras and zero divisors

I am going through a paper, https://www.sciencedirect.com/science/article/abs/pii/0196677488900144 and over there I found this statement in pg 2 confusing- The structure of associative algebra over $\...
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1answer
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Showing that $U(\mathfrak{sl}_2)$ is a coalgebra

We know that there is a coalgebra structure on $U(\mathfrak{sl}_2)$ as follows for any $z\in \mathfrak{sl}_2$: $$\Delta(z)=1\otimes z+z\otimes 1, \qquad \epsilon(z)=0.$$ Can someone be so kind to ...
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1answer
96 views

“NONCOMMUTATIVE Algebra with a view towards Algebraic Geometry”?

Is there a noncommutative algebra book that is similar to Eisenbud's "Commutative Algebra with a view towards Algebraic Geometry" in the sense that fundamental and geometrically motivated ...
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16 views

Idempotent Laurent polynomial (in noncommuting variables)

Let $K$ be a field and $R=K\langle X_1,\dots,X_n,X_1^{-1},\dots,X_n^{-1}\rangle$ the Laurent polynomial ring in $n$ noncommuting variables. Can $R$ have idempotents distinct from $0$ and $1$?
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1answer
64 views

Generalized Nakayama's lemma over a non-commutative ring

Let $R$ be a ring, $J(R)$ its Jacobson radical, $M$ is a finite $R$-module. The following statement is usually called Nakayama's lemma: if $IM=M$ then $M=0$. This is true over any ring (commutative or ...
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1answer
29 views

Can I always write group elements $L,R$ as $VA$ and $VA^{-1}$ respectively, for some $V,A$?

My first question is, given two different elements $L,R$ in the Lie group $SU(N)$ can one always make write $$L=V A, \quad \quad R=VA^\dagger,$$ for some $V, A \in SU(N)$ ? Secondly, how does this ...
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1answer
106 views

Commutativity of a prime ring

Question : Show that $R$ is a prime ring containing two commuting non-zero left ideals $I$ and $J$ $\implies$ $R$ is commutative, where "commuting ideals” means $ij=ji$ for all $i \in I$, $j \in ...
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44 views

Hardness of the conjugacy search problem

I have come across the following problem, which is considered as a mathematically hard problem to solve (one way trapdoor) used for cryptography. Conjugacy search problem: Let $G$ be a non-abelian ...
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1answer
61 views

Does a finitely generated faithful module over an Artinian ring contain a regular element?

In the text Nicholson -- Introduction to Abstract Algebra, 4th Ed (2012) the claim of exercise $8(b)$ of exercise set $11.1$ is: If $R$ is a left artinian ring with $1\ne 0$, and $M$ is a finitely ...
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27 views

On the Krull-Schmidt-Azumaya Theorem

$\newcommand\restr[2]{{% we make the whole thing an ordinary symbol \left.\kern{-\nulldelimiterspace} % automatically resize the bar with \right #1 % the function \vphantom{\big|} % pretend it's ...
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1answer
38 views

Higher commutators in rings

In Herstein's "Noncommutative rings" it is proved that any rng $R$ such that for every $x,y\in R$ there exists an integer $n(x,y)>1$ such that $$(xy-yx)^{n(x,y)}=(xy-yx)$$ must be ...
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49 views

A simple module isomorphism in Wedderburn theory

Given a semisimple ring $R$, we have a $R$-module isomorphism $R \cong n_1V_1 \oplus n_2V_2 \oplus\cdots \oplus n_lV_l$ where $V_1, V_2, \cdots, V_l$ are nonisomorphic simple $R$-modules. By ...
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1answer
41 views

Finding generalized inverses in the ring of linear transformations of a finite dimensional vector space V over a division ring

Let $L$ be the ring of linear transformations of a finite dimensional vector space $V$ over a division ring $D$. Show that for any $l$ belonging to $L$, then there exists a $u\in L$ such that $lul=l$. ...
3
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2answers
72 views

Why does the set of submodules of a module that are direct sums of irreducible submodules have a maximal element?

I was reading the proof that every R-module M of an artinian semisimple ring R is the direct sum of all irreducible R-submodules M. In the proof they stated that the set $$\mathcal{F}:=\{N\subset M: N ...
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1answer
41 views

explicit formula of $(x \frac{\partial}{\partial x})^n f(x)$

Is there any explicit expression of $$ (x \frac{\partial}{\partial x})^n f(x) $$ as function of $x$ and $\frac{\partial^{k}f}{\partial x^k} $$, $$ 1\leq k \leq n$. Any idea Thanks
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9 views

resolutions over a Leibniz algebra

Given the usual definitions of Leibniz algebra $L$, ideal $I$ and module $M$, has the theory of resolutions e.g. of $L/I$ over $L$, been worked out? citation please
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39 views

Is restriction of scalars preserving injective modules equivalent to flatness?

Given any ring homomorphism $R \to S$, if $S$ is a flat right $R$-module, then any injective left $S$-module is also injective as a left $R$-module. Now, I'm wondering whether the converse is true. ...
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1answer
26 views

Locally nilpotent derivations closed under isomorphism?

I have been struggling to prove or find a counterexample to the following: Fix a field $k$ of characteristic $0$. Let $R:= k[x_1,...,x_n]$ and $A$ be the ring of differential operators on $R$ (so $A$ ...
1
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1answer
53 views

Equality in non commutative field

Let $K$ be a non commutative field and $a,b \in K$ such that $ab \neq ba$. Show that : $$ a=\big[b-(a-1)^{-1}b(a-1)\big]\big[a^{-1}ba-(a-1)^{-1}b(a-1)\big]^{-1}. $$ I tried to move the second term of ...
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0answers
30 views

Jacobson radical and bimodules

I know that the Jacobson radical of a (non-commutative) rng $R$ is the intersection of all annihilators of right irreducible modules. I know also that one can replace "right" with "left&...
6
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1answer
41 views

An example about a non-commutative division ring with finite characteristic

After reading the proof of the theorem “For every central division $F$-algebra $D$ with $D$ $\neq$ $F$, $D$ contains a separable extension $K \supsetneqq F$“, I have a question: dose there exist a non-...
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38 views

Notation for functions of a Grassman variable

Let $V$ be a vector space over a field $\mathbb{K}$ and $\mathcal{O}(V)=\mathrm{Sym}V^*$ its ring of polynomial functions. Given $f_1,\dots,f_n\in V^*$, we have the homogeneous polynomial $F=f_1\cdots ...
2
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0answers
20 views

Fitting's Lemma version for pseudocompact modules or linearly compact modules

Let $R$ be a pseudocompact ring or a linearly compact ring. Is there a version of Fitting's Lemma for pseudocompoact or linearly compact $R$-modules?
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30 views

Taylor expansion for matrix exponential

Consider the matrix exponential $$U(x)=e^{ix^j T^j}$$ where $T^j$ are matrices (in my particular application $U\in SU(4)$ and $T^j$ are its generators) and $x^j \in \mathbb{R}$. I would like to know ...
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0answers
49 views

Can you give me some concrete examples of varietes of some algebras?

Can you give me some concrete examples of varieties of some algebras? I have seen many definitions for varietes of algebras, but I can not find concrete examples for this(for example, varietes of ...
4
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1answer
65 views

Infinite intersection of finitely generated ideals in a coherent ring.

It has been claimed without proof in several answers that an intersection of two finitely generated ideals in a coherent ring is finitely generated. Thus, the finitely generated ideals in a coherent ...
2
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2answers
72 views

Existence of Central Nilpotent Element implies that the Ring is not semi-simple

Show that if the ring $R$ with $1$ has a central nilpotent element then it is not semisimple. I couldn't find a solution directly but I have a solution. Since any central nilpotent element is ...
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1answer
37 views

Definition of commutative and non-commutative algebra and algebra isomorphism

I am not sure of the meaning of the notation C<<...>> used to define a commutative algebra A and non commutative algebra A^ in the image attached. I do understand the meaning of the ideal. ...
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0answers
10 views

Unimodular rows over a polynomial ring in two variables

Let $D$ be a division ring that is not a field and suppose $\alpha, \beta \in D$ with $\alpha\beta\neq \beta\alpha$. Show that $\sigma=(x+\alpha, y+\beta)$ is a unimodular row over $R=D[x,y]$. So I ...
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0answers
48 views

Example of a non-commutative ring containing 4 elements

I need an example of a non-commutative ring containing 4 elements. I considered the following example of $M_2(Z_2)$ containing the following four elements: $\begin{bmatrix}0&0\\0&0\end{bmatrix}...
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0answers
79 views

Constructing an isomorphism between vector spaces

Let $A$ be a central simple $k$-algebra of dimension $n^2$ over $k$. Also assume that $A= {\rm End} (V)$, where $V$ is an $n$-dimensional $k$-vector space. For $0\leq i \leq n$, let $M_i$ be the set ...
1
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1answer
58 views

What is the tensor product dependent on the field in an algebra?

I don't understand the following notation: $$V_F := V \otimes_k F$$ First of all, I know that that the product is a bilinear operation, i.e. $A \otimes A \to A$, between elements of the vector space $...
1
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1answer
48 views

An example of Artinian ring with an ideal K, where K is not Artinian ring?

It is true that an ideal of semisimple ring is semisimple ring. It is true that an ideal of an Artinian semisimple ring is an Artinian ring. What about just Artinian? Is there an example with the ...
3
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2answers
71 views

Morita equivalence and Brauer equivalence

Let $k$ be a field and $A,B$ be two (finite-dimensional) central simple $k$-algebras. We usually say that $A$ and $B$ are Brauer equivalent (or similar) if their underlying division algebras (given by ...

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