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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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1answer
111 views

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 ...
5
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0answers
47 views

Do twisted group rings of free abelian groups admit universal fields of fractions?

Let $R$ be an associative ring with unit. Recall that an epic $R$-field is a ring epimorphism $\alpha\colon R\to D$ to a skew field/division ring $D$. An epic $R$-field $\alpha$ is called a field of ...
5
votes
1answer
241 views

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 ...
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0answers
31 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 $...
4
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2answers
366 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 ...
18
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1answer
594 views

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\...
3
votes
1answer
127 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 ...
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191 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 ...
6
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1answer
174 views

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 ...
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0answers
61 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 ...
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2answers
334 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 $\...
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1answer
443 views

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 ...
3
votes
1answer
216 views

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?...
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1answer
85 views

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 $...
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0answers
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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,...
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1answer
117 views

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 ...
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1answer
286 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 ...
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1answer
527 views

Does there exist an infinite non-commutative division ring with finite center?

Does there exist an infinite non-commutative division ring with finite center?
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2answers
634 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 &...
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1answer
353 views

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 ...
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1answer
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(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-...
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2answers
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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) $V$ ...
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1answer
297 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 ...
18
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3answers
920 views

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 ...
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4answers
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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 ...
13
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2answers
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?
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5answers
2k views

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?