Questions tagged [noncommutative-algebra]

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

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3 votes
1 answer
28 views

Every order in a finite-dimensional $\mathbb{Q}$-algebra is contained in a maximal order

Definition. A lattice in a finite-dimensional $\mathbb{Q}$-algebra $V$ is a finitely generated $\mathbb{Z}$-submodule $\mathcal{L} \subset V$ such that $\mathcal{L}\mathbb{Q}=V$ (i.e., $\mathcal{L}$ ...
1 vote
0 answers
33 views

Orders in quaternion algebras

Definition. An algebra $B$ over a field $F$ is a quaternion algebra if there exists $i,j\in B$ such that $1,i,j,ij$ is a basis for $B$ as a vector space over $F$. Throughout, we fix $F=\mathbb{Q}$. ...
5 votes
1 answer
98 views

Examples of rings where every left ideal is two-sided but not every right ideal

Is there an example of a ring where every left-ideal is two-sided but not every right ideal? WHAT FOLLOWS IS A FAILED EXAMPLE As an argument but not a proof that the example fails, consider: $ f(a) \;...
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3 votes
1 answer
42 views

Ideals of the Lipschitz quaternions

Consider the subring $\mathcal{O}:=\mathbb{Z}+\mathbb{Z}i+\mathbb{Z}j+\mathbb{Z}k$ of the ring $\mathbb{H}=\mathbb{R}+\mathbb{R}i+\mathbb{R}j+\mathbb{R}k$ of real Hamiltonians. Let $I$ be a right ...
0 votes
0 answers
24 views

If $R$ is a product of matrix rings over division rings then every $R$ module is completely reducible

In Dummit and Foote's Abstract Algebra, the proof of the Wedderburn-Artin theorem is sketched out in a series of exercises for reader to solve. I'm currently working on the final exercise in the ...
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1 vote
1 answer
42 views

Every $R$-module is injective implies $R$ is a product of simple rings

In Dummit and Foote's Abstract Algebra, all the main results about representations of finite groups are derived from the Wedderburn-Artin theorem. The proof of the Wedderburn-Artin theorem itself is ...
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2 votes
1 answer
43 views

How to define the non-commutative ring $\mathbb{F}_{4}+e\mathbb{F}_{4}$, $e^2=1$, $ae=ea^2$ in MAGMA(Computational Algebra System)?

I'm trying to learn to use MAGMA(Computational Algebra System) for research in coding theory over non-commutative rings, but it's been slow going. I feel like it's hard to find anything in the ...
4 votes
0 answers
58 views

Commuting operators vs. commutative rings, and localization in the quantum mechanical sense vs. commutative algebra sense.

One of the answers in this MO post https://mathoverflow.net/questions/7917/non-commutative-algebraic-geometry draws a connection between localization in commutative algebra (or the failure thereof), ...
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1 vote
0 answers
14 views

Power function in non-commutative algebras

Is there a canonical form of the power function, $a^b=x$, that extends to non-commutative algebras like matrices and hypercomplex numbers? It is known that $a^b=e^{b\log{a}}$ for commutative algebras, ...
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1 vote
1 answer
64 views

Spectral triple on $\mathcal C(M)$ where $M$ is a compact Riemannian manifold, not necessarily spin

I have been reading Alain Connes' Compact metric spaces, Fredholm modules and hyperfiniteness. In proposition 1, it is mentioned that an unbounded Fredholm module (nowadays: spectral triple) over $C(M)...
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0 votes
2 answers
74 views

Meaning of "free on the basis"

T.Y. Lam in his book "A First Course in Noncommutative Rings" gave an example of "Hurwitz ring of integral quaternions" which is $$R=\lbrace(a+bi+cj+dk)/2 \mid a,b,c,d\in\mathbb{Z} ...
0 votes
1 answer
39 views

Example of a quasi nilpotent element which is not a nilpotent element

Let $R$ be a ring with unity. An element $a\in R$ is said to be a quasi nilpotent element of $R$ if $1-ax$ is unit for all $x\in comm(a)$ where $comm(a)=\lbrace x \in R | ax=xa\rbrace $. It is obvious ...
0 votes
2 answers
55 views

Spectral triple for a (real) full matrix algebra

Let $\mathcal A = \mathbb R^{N \times N}$ be the real full matrix algebra, $N \in \mathbb N_{> 1}$, which is represented by the Hilbert space $H := \mathbb R^N$ (that is, $\mathcal A \to B(H)$, $A \...
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1 vote
2 answers
55 views

Is there are a ring where $\frac{1}{2}$ doesn't commute with everything?

Is there a ring with an element $x$ such that $2x=1$ which is non-central (there is some $y$ such that $xy \neq yx$)? Suppose there is, then by taking the subring generated by $x$ and $y$, there had ...
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2 votes
1 answer
39 views

Noncommutative extension of algebra of smooth functions

Let $A$ denote the algebra of smooth real functions on $\mathbb{R}^n$. Let $j : \mathbb{R}[x_1,\ldots,x_n] \rightarrow A$ be the injective homomorphism sending a polynomial to its corresponding ...
1 vote
1 answer
28 views

Set of all elements of a ring which semicommute with other elements of a ring

I am wondering that as we define $Z(R)$, center of a ring is a subring of $R$. Can we define a subset which is collection of all those elements which semicommute with other elements. Will this set ...
5 votes
1 answer
63 views

Central simple algebras and semisimple subalgebras

Let $B$ be a central simple algebra over an algebraically closed field $k$, and let $A\subset B$ be a semi-simple $k$-subalgebra. By this I also mean $k$ is central in $A$ as well, and the map $k\to A\...
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2 votes
1 answer
87 views

The C$^*$ algebras of matrices, continuous functions, measures and matrix-valued measures / continuous functions and their state spaces

I have no background in Algebra, but want to understand matrix-valued measures and matrix-valued continuous functions from the C$^*$-algebra perspective to identify what definition of matrix-valued ...
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0 votes
0 answers
31 views

Either $r$ or $1-r$ is left-invertible in a ring $R$ with $1$ $\iff$ either $r$ or $1-r$ is a unit.

Let $R$ be a noncommutative ring with $1$. I need to show that for any $r\in R$ either $r$ or $1-r$ is left-invertible if and only if either $r$ or $1-r$ is a unit. Proof. Indeed, if either $r$ or $1-...
0 votes
1 answer
21 views

Every non-unit in a non-commutative ring with unity is contained in a maximal left ideal.

Suppose that $x$ is a non-unit in a non-commutative ring $R$ with unity $1$. Is it true that $x$ is contained in some maximal left ideal of $R$. I tried to prove the statement by Zorn's Lemma by ...
2 votes
2 answers
110 views

Fredholm module over $\mathbb R^{N \times N}$ and over matrix-valued measures

The following is taken from Compact metric spaces, Fredholm modules, and hyperfiniteness by Alain Connes, cf. also his paper Non-commutative differential geometry. Definition. Let $A$ be a unital $C^*$...
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1 vote
1 answer
47 views

Semi-commutative but not commutative ring.

So, P.P Nielsen created an example of a semicommutative ring which is not McCoy (or say commutative). He took, $k=\mathbb{F}_2\left\langle a_0,a_1,a_2,a_3,b_0,b_1\right\rangle$ be the free algebra (...
0 votes
0 answers
50 views

(local) Artinian rings with nilpotent Jacobson Radical and Maximal ideals

Problem 1: If $R$ is an Artinian ring, then show that the Jacobson Radical $J(R)$ is nilpotent. Problem 2: If $(R, \mathfrak{m})$ is a local Artinian ring, then show that $\mathfrak{m}$ is nilpotent. ...
1 vote
1 answer
45 views

Let $A$ be a central division algebra (of finite dimension) over a field $k$. Show that $[A,A] \neq A$.

I am looking at the post A central division algebra is not its commutator and I have a few questions regarding the proof that was provided in the answer. Why is $A$ a simple $k$-algebra? My first ...
2 votes
1 answer
50 views

Are Weyl algebra $A_1$ and it's opposite algebra isomorphic?

Let $A$ be a noncommutative ring and $ab=c$ in $A$. $A'$ is it's opposite ring if $ba=c$ in $A'$. If $A$ is a Weyl algebra $A_1$, are $A$ and $A'$ isomorphic? I have an idea, but I think it's wrong. ...
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3 votes
1 answer
36 views

Uniqueness of a submodule

Suppose I want to show that $\mathbb{C}[X]$ is the unique simple $A_1$-submodule of $\mathbb{C}[X,X^{-1}]$, where $A_1 = A_1(\mathbb{C})$ is the first Weyl Algebra. It is not difficult to show it is ...
  • 115
0 votes
1 answer
42 views

quintic solution to noncommutative polynomial

So, we know there are generally no solutions to an arbitrary quintic polynomial (EDIT from comments: Of course, I mean you can't write the roots using radicals, not that the roots don't exist). I don'...
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0 votes
0 answers
47 views

Computing the nilradical of a ring

Let $R=\begin{pmatrix} \mathbb{C} & \mathbb{C} & \mathbb{C}\\ 0 & \mathbb{C} & \mathbb{C}\\ 0 & \mathbb{C} & \mathbb{C} \end{pmatrix}$. I want to find the nilradical: $$N(R)=\...
  • 115
2 votes
1 answer
37 views

Product of linearly dependent vectors is 0 in an anti-commutative algebra

From the 3rd edition of the book "The Linear Algebra a Beginning Graduate Student Ought to Know" by Jonathan S. Golan, we find the following statement in chapter 4: Note that if ${v,w}$ is ...
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0 votes
0 answers
45 views

How to determine if a non-commutative algebra is semisimple

Let $\cal{Q}$ be the field of rationals and $L=\cal{Q}(\sqrt 2, \sqrt 3)$ be a Galois extension of degree 4 ($[L:\cal{Q}]=4$). Using theory of Drinfeld twists on the galois group of $L$, I have ...
2 votes
1 answer
89 views

A division quaternion algebra in which the integral elements don't form a ring

I wish to find a division quaternion algebra $B$ over $\mathbb{Q}$ and elements $\alpha, \beta\in B$ such that $\alpha, \beta$ are integral over $\mathbb{Z}$ but both $\alpha + \beta$ and $\alpha \...
3 votes
1 answer
137 views

Prove: $P(XY=YX) \leq 5/8$ in a finite no commutative group

Question: Let $(G;.)$ a finite no commutative group. Find that an upper bound that to the probability that two element $X,Y$ of $G$ randomly chosen with an uniforme probability (that means the ...
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0 votes
1 answer
49 views

How is invertibility defined for square matrices over non-commutative rings?

In this article Wikipedia defines invertibility for square matrices over commutative rings as follows: ...in the case of the ring being commutative, the condition for a square matrix to be invertible ...
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0 votes
0 answers
33 views

Reference of automorphism with respect to locally nilpotent derivations

Let $ R $ be a commutative algebra. Let $ \delta $ be a locally nilpotent derivation, that is, there exists some $ n $ such that $ \delta^{n} = 0 $. Then we can define a automorphism, namely $ \exp(...
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3 votes
2 answers
149 views

Why do we say that the tensor product of vector spaces is commutative, but the tensor product of vectors is not?

The Wikipedia article on the tensor product says The tensor product of two vector spaces $V$ and $W$ is commutative in the sense that there is a canonical isomorphism V ⊗ W ≅ W ⊗ V that maps v ⊗ w to ...
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0 votes
1 answer
47 views

Definition of a basic algebra over a field $K$

Let $K$ be an algebraically closed field and let $A$ be a $K$-algebra with a complete set $\{e_1,…,e_n\}$ of primitive orthogonal idempotents. So, the algebra $A$ is called basic if $$e_iA \cong e_jA \...
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1 vote
0 answers
20 views

Seeing that a product of quasi-free algebras is quasi-free in terms of lifting homomorphisms

A $k$-algebra over a field $k$ is quasi-free if for any bimodule $M$ the second Hochschild cohomology $H^2(A,M)$ vanishes or, equivalently, if for any square-zero extension $R/M=A$ there is a lifting ...
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1 vote
0 answers
37 views

A generalization of the Clifford algebra

A minimal example of the Clifford algebra is the $\mathbb{C}$-algebra (unital, associative) generated by $x,y$ quotient over the relations \begin{eqnarray} x^2&=&1,\tag{1}\\ y^2&=&1,\...
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0 votes
1 answer
28 views

Proof of (left) Ore condition implies there exists Q(R) a left quotient ring of R.

I am trying to understand the proof of Theorem 7.1.1 of `Noncommutative Rings' I. N. Herstein. Definition: An element in a ring is said to be regular if it is neither a left nor a right zero divisor ...
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1 vote
0 answers
31 views

Request for reference books, articles, papers on the topic topic of Non Abelian Kummer Extensions.

I am a final year undergraduate student. I am doing my research on Non-Abelian Kummer Extensions. Can someone please introduce me to formal definition of Non-abelian Kummer Extensions. I know it is ...
1 vote
0 answers
33 views

Help me Understanding Non Abelian Kummer Extensions

[![I am doing my undergraduate research work on the Non-Abelian Kummer Extensions. I am following the book "Algebra" by Serge Lange. I have understood the difference between the abelian ...
1 vote
1 answer
32 views

Does Mashke's theorem hold for semisimple rings?

I saw a proof of Mashke's theorem using the theory of modules. However, it seems like it works in much more generality than it was stated in the text; can anyone confirm if it does, or if there's ...
user avatar
1 vote
0 answers
24 views

Number of solutions to simultaneous equations on lie groups

Say I have $n$ variables, special unitary operators from $SU(k)$, and write a set of $m$ equations that they must satisfy. These have the form $UVW...Z=I$, i.e. each one specifies that some product of ...
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1 vote
1 answer
35 views

If E is a splitting field for A then it is clear that there exists a finitely generated subfield E'/F that is also a splitting field. [closed]

I saw in Jacobson's book Finite Dimensional Division Algebras (1996) page 158 the sentence "If E is a splitting field for A then it is clear that there exists a finitely generated subfield E'/F ...
2 votes
1 answer
84 views

Necessary and sufficient conditions for the function $f(t) := \mbox{trace}(A^{-1}e^{-tB} A e^{-tB})$ to be monotone decreasing.

Let $A$ and $B$ be a positive-definite $n \times n$ matrices. For any $t \ge 0$, define $f(t) := \mbox{trace}(A^{-1}e^{-tB} A e^{-tB}) = \|A^{-1/2}e^{-tB}A^{1/2}\|_F^2$. Question. What are necessary ...
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3 votes
1 answer
56 views

division rings $D,D'$ both of finite dimension over their center $F$ are isomorphic as rings iff isomorphic as $F$-algebras?

Say I have two division rings, $D,D'$, both with center $F$ and both are of finite dimension over $F$, for some field $F$. Now suppose that $D\cong D'$ as rings, does if follow that $D\cong D'$ as $F$ ...
3 votes
1 answer
90 views

Nakayama’s Lemma gives $\mathrm{rad}(S) \subset \mathrm{rad}(R)$ for $S \subset R$ under certain finiteness conditions

I'm preparing for the qualify exam and coming up with the following exercise: Question: Let $S \subset R$ be a subring contained in the center of $R$. Suppose that $R$ is finitely generated as a left $...
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0 votes
0 answers
31 views

Subalgebras which are semisimple

Let $k$ be a field. Let $A$ be a $k$-algebra and $B \subset A$ be a subalgebra of $A$ which is a semisimple algebra. Note that we do not assume that $A$ is commutative. We also assume that $A$ is a ...
2 votes
0 answers
67 views

Jacobson density theorem, and its relations to Artin-Wedderburn, and double centralizer theorems

$\newcommand{\End}{\operatorname{End}}\newcommand{\Hom}{\operatorname{Hom}}$ On pg. 647 of Lang's Algebra, Lang proves the Jacobson density theorem by doing some stuff with $\End_{\End_R(V^n)}(V^n)$ ...
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1 vote
0 answers
71 views

I've proven Krull-Schmidt for arbitrary decomposition into indecomposables. What's wrong in my proof?

My question is probably stupid and I'm likely committing a very trivial mistake. It's well known that the uniqueness of decomposition of modules into indecomposable submodules ${}_A M = \bigoplus_{I \...
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