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### Division ring and vector space [duplicate]

How one can define right vector space over Division ring $R$ ? What properties it lack than the vector space over field? One thing which it lack is definitely commutative properties but what other ...
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### What is the motivation for quaternions?

I know imaginary numbers solve $x^2 +1=0$, but what is the motivation for quaternions?
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### An example of a division ring $D$ that is **not** isomorphic to its opposite ring

I recall reading in an abstract algebra text two years ago (when I had the pleasure to learn this beautiful subject) that there exists a division ring $D$ that is not isomorphic to its opposite ring. ...
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### Do people ever study non-commutative fields?

I've heard of a field, and I've heard of a non-commutative (or "not-necessarily commutative) rings. Do people ever study non-commutative fields?
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### What changes for linear algebra over a finite field?

This question asks which standard results from linear algebra over a field no longer hold when we generalize the algebraic structure of the scalars to be an arbitrary division ring. My question is ...
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### An example of noncommutative division algebra over $Q$ other than quaternion algebras

Could anyone please show me an example of finite dimensional noncommutative associative division algebra over the field of rational numbers $Q$ other than quaternion algebras?
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### Nilpotent matrix over a division algebra

Suppose I have an $n\times n$ nilpotent matrix $A$. If the entries are from any field, then I can show that all eigenvalues are zero and the trace is zero. Indeed, if we consider the algebraic closure ...
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### What's bad about left $\mathbb{H}$-modules?

Can you give me non-trivial examples of propositions that can be formulated for every left $k$-module, hold whenever $k$ is a field, but do not hold when $k = \mathbb{H}$ or, more generally, need not ...
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### Every module over a division ring is free?

I am currently trying to answer the following true/false question: True or False: Every module over a division ring $R$ is free. I know the result would be true if $R$ is a field (ie a ...
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### How does linear algebra over the octonions and other division algebras work?

An interesting question, which has been discussed in many forms on this site, is how many results from the study of linear algebra over vector spaces carries over when we allow the scalars to form an ...
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