# 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
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### 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 ...
0answers
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### 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?
1answer
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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 ...
0answers
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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 ...
1answer
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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 ...
1answer
312 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 ...
1answer
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2answers
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### 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?
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?