# 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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### 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 $k$-...
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### 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 $R$ ...
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### 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 $D'$ ...
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### Multiplicative groups of skew fields

Is every group isomorphic to a subgroup of the multiplicative group of some skew field?
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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 ...
131 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?
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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 ...
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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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### 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 ...
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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 ...
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