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Questions tagged [division-ring]

Use this tag for questions about division rings in abstract algebra and/or noncommutative algebra.

In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a nonzero ring in which every nonzero element a has a multiplicative inverse, i.e., an element $x$ with $a·x=x·a=1$. A division ring is a type of noncommutative ring.

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If $D$ be a division ring and $D^*$ be finitely generated group then $D^*$ is abelian group?

Wedderburn's little theorem : every finite division ring $D$ is commutative, or $D^*$ is abelian group. Now if $D^*$ be a finitely generated group then $D^*$ is an abelian group ?
amir bahadory
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Abelianisation of a division ring

I was reading a paper recently concerning a non-commutative version of the matrix determinant. On the third page, it stated a fact without providing a proof or a reference: If $D$ is a division ring, let $D^\times$ be its multiplicative group, then…
AdJoint-rep
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Galois theory for (non-commutative) division rings

Is there a 'Galois theory' with fields replaced by (non-commutative) division rings? I have googled this, and it seems that there are known results in that direction, for example, this paper which says: "Galois theory was extended to division…
user237522
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If $(\mathbb R,<,+,\boxdot)$ is an ordered division ring, is it automatically a field?

I have an operation $\boxdot$ over $\mathbb R$ which makes $(\mathbb R,<,+,\boxdot)$ into an ordered division ring (field without commutativity for $\boxdot$, or rather such commutativity is not proven yet). Of course, $\boxdot$ is compatible with…
Enrico
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Proving Division Rings of $p^2$ Elements are Fields

Exercise III.2.11 (Aluffi, Algebra Ch 0): Let $R$ be a division ring consisting of $p^2$ elements, where $p$ is a prime. Prove that $R$ is commutative (and thus $R$ is a field). Note: I do so without invoking Wedderburn's Theorem, that every…
BabyDietCoke
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Reflection groups of division rings

My question is: Is there a classification of finite groups representable as a $\mathbb{K}$-reflection group for some division ring $\mathbb{K}$ of characteristic zero? I would also appreciate any references dealing with this problem or with…
pregunton
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What is the intuition behind Hua’s proof of the Cartan-Brauer-Hua theorem?

The Cartan-Brauer-Hua theorem states that Let $K\subset D$ be division rings so that whenever $x\in D$ is a nonzero element, $xKx^{-1}\subset K$. Show that either $K\subset Z(D)$ or $K=D$. This theorem is partially named after Hua due to his…
BAI
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A finite division ring $D$ is a field

7.8.12 Wedderburn: A finite division ring $D$ is a field. I have understood several theorems from this book (A first course in abstract algebra by Hiram, paley) by my own, but this one is beyond everything. Every time I give a try to understand this…
N00BMaster
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Definition of a division ring in category theory

I'm wondering how one can define a division ring in category theory. More precisely, is there a well-defined concept of "division ring object" such that a division ring object in the category of sets is a division ring in the standard sense ? The…
Sephi
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Proof of brauer's lemma, $eRe$ being a division ring.

On page 1 of this article, the author proves the following claim: Brauer's Lemma: Let $K$ be a minimal left ideal of a ring $R$, with $K^2 \not= 0$. Then $K=Re$ where $e^2=e \in R$, and $eRe$ is a division ring. What I do not understand is why…
Bryan Shih
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The division ring of fractions of the first Weyl algebra and its subrings

The first Weyl algebra, $A_1(k)= k\langle x,y | yx-xy=1\rangle$, where $k$ is a field of characteristic zero, is known to be a simple Noetherian ring, hence it has a (left) division ring of fractions (see this question), denote it by $D_{x,y}$. Let…
user237522
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The multiplicative group of the real quaternion division ring

Let $\mathbb{H}$ be the real quaternion division ring, that is, $\mathbb{H}$ consists of all elements of the form: $a+bi+cj+dk$ in which $a,b,c,d\in\mathbb{R}$ and $i^2=j^2=k^2=-1,ij=-ji=k$ with usual addition and multiplication. Then, the set…
Tran Nam Son
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When is $R/M$ a division ring?

There is a famous theorem in commutative ring theory which states: "Let $R$ be a commutative ring with unity and let $M$ be a (two-sided) ideal of $R$. Then, $M$ is maximal if and only if $R/M$ is a field". This is not valid if $R$ is noncommutative…
Gauss
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An example of a ring which is very close to division ring but not a division ring.

Let $R$ be a ring with unity. An element $a\in R$ is said to be a unit element if there exists $b\in R$ such that $ab=ba=1$. The ring $R$ is called a division ring if every nonzero element is a unit element. An element $f\in R$ is said to be a full…
Dr. Nirbhay Kumar
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Why are real numbers on the number line but complex numbers aren't?

This seems like a question that should be relatively easy to answer, but for the life of me I simply can't figure it out. My question is relatively simply put in the title, and the answer seems like it should be that $\mathbb{R}$ is a one…
tox123
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