Questions tagged [division-rings]

A division ring is a possibly noncommutative ring where every nonzero element has a two-sided multiplicative inverse.

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Is there a classification of reflection groups over division rings?

I asked a version of this question in Math StackExchange about a week ago but I've received no feedback so far, so following the advice I received on meta I decided to post it here. Details The ...
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Can one embed two division rings in a common one?

I could not get an answer to this question in MathStackExchange, so I dare ask it here. Given any two fields, $\rm F_1,F_2$ over the same prime subfield $\rm F$, the quotient $\rm \mathbf F=F_1\...
Drike's user avatar
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18 votes
3 answers
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units in distinct division algebras over number fields---are they definitely not isomorphic as abstract groups?

This is really an irrelevant question in the sense that the answer isn't remotely "logically crucial for the Langlands programme" or whatever---it's just something that occurred to me when writing ...
Kevin Buzzard's user avatar
17 votes
4 answers
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Dimension of central simple algebra over a global field "built using class field theory".

If $F$ is a global field then a standard exact sequence relating the Brauer groups of $F$ and its completions is the following: $$0\to Br(F)\to\oplus_v Br(F_v)\to\mathbf{Q}/\mathbf{Z}\to 0.$$ The ...
Kevin Buzzard's user avatar
14 votes
2 answers
1k views

Free division rings?

Does it make sense to talk about, say, the free division ring on 2 generators? If so, does the free division ring on countably many generators embed into the free division ring on two generators?
Uri Andrews's user avatar
8 votes
5 answers
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Proof a Weyl Algebra isn't isomorphic to a matrix ring over a division ring

Can anyone prove that a Weyl Algebra is not isomorphic to a matrix ring over a division ring?
Casebash's user avatar
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7 votes
2 answers
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Bimodules over division rings

Inspired by other questions i have two questions about modules over division rings: given a division ring $D$ with center $Z(D)=K$. One has the notion of dimension for left modules (vector spaces) $V$ ...
TonyS's user avatar
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7 votes
1 answer
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What does $K_1(R)$ tell us about $GL_n(R)/E_n(R)$?

Let $D$ be a division ring, and $R=D[t_1,\ldots,t_n]$. If $GL_m(R)$ is the usual group of invertible matrices over $R$, then by $E_m(R)$ I mean the subgroup of $GL_m(R)$ generated by the elementary ...
Sam Williams's user avatar
6 votes
1 answer
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Does there exist an infinite non-commutative division ring with finite center?

Does there exist an infinite non-commutative division ring with finite center?
Mamiri's user avatar
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6 votes
0 answers
437 views

Algebraic-closures of division rings

In what follows, $x$ is always taken to commute with the coefficient ring. This means that for any given polynomial, you can put the coefficients to the right or the left of $x$ as you please. This ...
Jonathan Gleason's user avatar
5 votes
1 answer
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Multiplicative groups of skew fields

Is every group isomorphic to a subgroup of the multiplicative group of some skew field?
Daniel Sebald's user avatar
5 votes
1 answer
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Counterexample for the Skolem-Noether Theorem

If a division ring is finite-dimensional over its center then we can apply Skolem-Noether theorem (which asserts that every endomorphism is a conjugation). Can someone give a counterexample of the ...
user15749's user avatar
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4 votes
1 answer
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Division ring on a field

Suppose that $F$ is a field. Show that there exists a $F$-division algebra $D$ with two elements $a\neq b\in D$ such that $a^2-2ab+b^2=0$. In the field extensions we know that $a^2-2ab+b^2=0$ if and ...
MH.Fakharan's user avatar
4 votes
2 answers
624 views

Modules over infinite rings which can not be a finite union of their proper submodules

It is well known that a vector space over an infinite field cannot be a finite union of its proper subspaces. Does this fact have an immediate and obvious generalization to modules over infinite ...
Ali Taghavi's user avatar
4 votes
1 answer
172 views

Subfields of division rings of degree $2$ which are not invariant

Let $A$ be a noncommutative division ring, and let $B$ be a sub division ring (here, $B$ is allowed to be commutative) of degree $2$. Are there easy examples known for which $B$ is not globally fixed ...
THC's user avatar
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4 votes
1 answer
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Analysis in Division Rings

In the question here, the subject of "Analysis in Positive Characteristic" is mentioned. Looking at Wikipedia's local field, this is the final type of analysis in local fields to be developed ...
user19172's user avatar
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3 votes
1 answer
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Elementary classification of division rings

Are there examples (other than the two mentioned below) of fields $K$ such that the classification of all finite dimensional division $K$-algebras is possible using only elementary theory (lets say a ...
Mare's user avatar
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3 votes
1 answer
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Infinite dimensional finitely generated algebraic division algebra

Is there a division algebra $D$ with center $K$ that satisfies the following 3 conditions? 1) $D$ is of infinite dimension over $K$; 2) every element of $D$ is algebraic over $K$; 3) $D$ is ...
Andrei Jaikin's user avatar
3 votes
1 answer
110 views

Dimension of division rings coming from indecomposable modules

Let $k$ be a field, $A$ a $k$-algebra and $X$ a finite dimensional indecomposable $A$-module. Then $\text{End}_A (X)$ is a local ring. Let $m$ be its maximal ideal. Can we say anything about the $k$-...
kevkev1695's user avatar
3 votes
1 answer
196 views

Extending an automorphism to an inner one

Let $D$ be a division ring. I have in mind the following result. Theorem. For every automorphism $f$ of $D$, there is a division ring $E$ extending $D$ such that $f$ extends to an inner automorphism ...
Drike's user avatar
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3 votes
1 answer
117 views

Charaterisation of quaternion algebras

Let $k$ be a field, and $A$ an associative $k$-algebra with an identity element. Say that $A$ is quadratic if any subalgebra of $A$ generated by a single element has dimension at most two. I am ...
Erik D's user avatar
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3 votes
1 answer
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Galois extensions inside a division ring

Let $D$ be a division ring which has finite dimension over its centre. Q1. Under which conditions can one find a maximal subfield $K$ of $D$ and a proper subfield $L$ of $K$ such that $K/L$ is Galois?...
Drike's user avatar
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3 votes
1 answer
203 views

Fractional ideals of maximal orders in quaternion algebras

Let D be a skew field that is central and finite-dimensional over a number field F (in particular: a quaternion algebra over F). Let $\Delta$ $\subseteq$ D be a maximal $\mathcal{O}$$_{F}$-order. Let $...
jgerrit's user avatar
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1 answer
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For a division ring $D$, does $[D:C_D(a)]_{right}$ vary when $D$ is enlarged?

In a commutative field $K$, the Zariski dimension of an algebraic subset of $K^n$ over $K$ does not vary if one enlarges $K$ if I understood well. In particular, for two Zariski-closed vector spaces $...
Drike's user avatar
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2 votes
1 answer
268 views

(Non-)existence of skew fields satisfying a SGPI (=skew generalized polynomial identity)

Let $K$ be a skew-field, infinite dimensional over its center $F$. From Kaplansky's PI-theorem it then follows that $K$ cannot satisfy a polynomial identity (the theorem says that primitive PI-...
Max Horn's user avatar
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2 votes
0 answers
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Cross-ratio for projective lines over division rings

If one considers a projective line over a field $k$, then the cross-ratio $(w,x;y,z)$ is a well-known geometric tool. But what if $k$ is not commutative, that is, if $k$ is a division ring ? Is there ...
THC's user avatar
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2 votes
0 answers
102 views

Product of two involutions in $\mathrm{PSL}_2(D)$

Let $D$ be a division ring and $\mathrm{PSL}_2(D)$. Suppose that $\overline{A}\in\mathrm{PSL}_2(D)$ where $A\in \mathrm{SL}_2(D)$. If $\overline{A}$ is identity, then $\overline{A}$ can express two ...
Tran Nam Son's user avatar
2 votes
0 answers
153 views

Division rings with finitely generated group of units

Is there any classification of division rings with finitely generated group of units? Is there any non-trivial example?
Sh.M1972's user avatar
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1 vote
1 answer
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Semisimple elements in division algebras

I found the following exercise at page 85 of the Strade-Farnsteiner's book "Modular Lie algebras and their representation": Let $D$ be a finite-dimensional division ring over a field $F$ of ...
Rocky Smith's user avatar
1 vote
1 answer
314 views

Are the elements of a division algebra which commute with all commutators in the center of the algebra?

I asked this quetion five days ago at https://math.stackexchange.com/questions/406669/are-the-elements-of-a-division-algebra-which-commute-with-all-commutators-in-the Some good people have given good ...
yanyu's user avatar
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1 vote
2 answers
961 views

What structure supports division to a unique quotient and remainder?

This has been bugging me for a while. According to https://en.wikipedia.org/wiki/Euclidean_division, if I divide integer $a$ by integer $b$, I get unique $t$, $r$ such that $a = t b + r$, $0 \le r &...
Benjamin Braun's user avatar
1 vote
0 answers
53 views

Commutator length of the center $Z(D')$ of $D'$ in a division ring $D$

Let $D$ be a division ring, and $D^\times$ the multiplicative group of $D$. Denoted $D'$ (resp. $Z(D')$) by the derived subgroup of $D^\times$ (resp. the center of $D'$). Here, we consider $D'$ ...
Tran Nam Son's user avatar
1 vote
0 answers
65 views

There is a ring with multiplication. Can we find a formula for division based on formula for multiplication?

Studying divergent integrals, I found a good formula for their multiplication: $\int_0^\infty f(x)dx\cdot\int_0^\infty g(x)dx=\int_0^\infty D^2 \Delta^{-1} \left(\Delta D^{-2}f(x)\cdot\Delta D^{-2}g(x)...
Anixx's user avatar
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1 vote
0 answers
63 views

Is left dimension preserved by left translation?

Let ${\bf K}\supset K\supset L$ be division rings with $[K:L]_{\rm left}=\infty$, and $a\in {\bf K}^\times$. Question. Is it possible that $[aK:L]_{\rm left}<\infty$ in the sense that there would ...
Drike's user avatar
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1 vote
0 answers
82 views

If $A$ is an integer ring such that each $P \in A_L[X]$ has a finite number of zeros in $A$, is $A$ commutative?

Let $A$ be a ring in which the product of any two nonzero elements is nonzero (we shall say that $A$ is an integral domain, even if $A$ is non commutative). It is well-known that if $A$ is commutative,...
Stabilo's user avatar
  • 1,419
0 votes
2 answers
360 views

Counting Divisors in $\mathbb{Z}^n$

Basically, I'm looking for ways to multiply elements of $\mathbb{R}^n$ that allow me to count divisors in $\mathbb{Z}^n$. For every positive integer $n$, I'm looking for an algebra structure on $\...
Linden's user avatar
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0 votes
1 answer
322 views

Uniqueness of maximal subfields

Let D be a division ring with center Z. Let R and K be two maximal subfields of D, both purely inseparable of exponent one ( means the p power of each of them in Z). Why are R and K isomorphic? Or a ...
user8321's user avatar
0 votes
0 answers
73 views

Squares in division ring extensions $\ell/k$ with $[\ell:k] = 2$

Let $k$ and $\ell$ be division rings such that $\ell$ contains $k$, and $[\ell : k] = 2$. When do I know that there is an element $a \in k$ such that $x^2 = a$ has solutions in $\ell$, but not in $k$?
THC's user avatar
  • 4,025
0 votes
1 answer
115 views

Full matrix ring over an infinite division ring with a finite maximal unital subring?

I'm wondering if there is an infinite division ring $D$ and a finite unital subring $R$ of the full matrix ring $M_n(D)$ ($n$ some positive integer) such that there are no rings properly between $R$ ...
Greg's user avatar
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