When I was studying linear algebra in the first year, from what I remember, vector spaces were always defined over a field, which was in every single concrete example equal to either $\mathbb{R}$ or $\mathbb{C}$.
In Associative Algebra course, we sometimes mentioned (when talking about $R$-modules) that if $R$ is a division ring, everything becomes trivial and known from linear algebra.
During the summer, I'm planning to revisit my notes of linear algebra, write them in tex, and try to prove as much as possible in a general setting.
Are there any theorems in linear algebra, that hold for vector spaces over a field and not over a division ring? How much linear algebra can be done over a division ring?
Also, what are some examples of division rings, that aren't fields? $\mathbb{H}$ is the only one that comes to mind. I know there aren't any finite ones (Wedderburn). Of course, I'm looking for reasonably nice ones…
Best Answer
In my experience, when working over a division ring $D$, the main thing you have to be careful of is the distinction between $D$ and $D^{op}$.
E.g. if $F$ is a field, then $End_F(F) = F$ ($F$ is the ring of $F$-linear endomorphisms of itself, just via multiplication), and hence $End(F^n) = M_n(F)$; and this latter isomorphism is what links matrices and the theory of linear transformations.
But, for a general division ring $D$, the action of $D$ by left multiplication on itself is not $D$-linear, if $D$ is not commutative. Instead, the action of $D^{op}$ on $D$ via right multiplication is $D$-linear, and so we find that $End_D(D) = D^{op}$, and hence that $End_D(D^n) = M_n(D^{op}).$
As for examples of division algebras, they come from fields with non-trivial Brauer groups, although this may not help particularly with concrete examples.
A standard way to construct examples of central simple algebra over a field $F$ is via a crossed product. (Unfortunately, there does not seem to be a wikipedia entry on this topic.)
What you do is you take an element $a\in F^{\times}/(F^{\times})^n$, and a cyclic extension $K/F$, with Galois group generated by an element $\sigma$ of order $n$, and then define a degree $n^2$ central simple algebra $A$ over $F$ as follows:
$A$ is obtained from $K$ by adjoining a non-commuting, non-zero element $x$, which satisfies the conditions
This will sometimes produce division algebras.
E.g. if we take $F = \mathbb R$, $K = \mathbb C$, $a = -1$, and $\sigma =$ complex conjugation, then $A$ will be $\mathbb H$, the Hamilton quaternions.
E.g. if we take $F = \mathbb Q_p$ (the $p$-adic numbers for some prime $p$), we take $K =$ the unique unramified extension of $\mathbb Q_p$ of degree $n$, take $\sigma$ to be the Frobenius automorphism of $K$, and take $a = p^i$ for some $i \in \{1,\ldots,n-1\}$ coprime to $n$, then we get a central simple division algebra over $\mathbb Q_p$, which is called the division algebra over $\mathbb Q_p$ of invariant $i/n$ (or perhaps $-i/n$, depending on your conventions).
E.g. if we take $F = \mathbb Q$, $K =$ the unique cubic subextension of $\mathbb Q$ contained in $\mathbb Q(\zeta_7)$, and $a = 2$, then we will get a central simple division algebra of degree $9$ over $\mathbb Q$. (To see that it is really a division algebra, one can extend scalars to $\mathbb Q_2$, where it becomes a special case of the preceding construction.)
See Jyrki Lahtonen's answer to this question, as well as Jyrki's answer here, for some more detailed examples of this construction. (Note that a key condition for getting a division algebra is that the element $a$ not be norm from the extension $K$.)
Added: As the OP remarks in a comment below, it doesn't seem to be so easy to find non-commutative division rings. Firstly, perhaps this shouldn't be so surprising, since there was quite a gap (centuries!) between the discovery of complex numbers and Hamilton's discovery of quaternions, suggesting that the latter are not so easily found.
Secondly, one easy way to make interesting but tractable non-commutative rings is to form group rings of non-commutative finite groups, and if you do this over e.g. $\mathbb Q$, you can find interesting division rings inside them. The one problem with this is that a group ring of a non-trivial group is never itself a division ring; you need to use Artin--Wedderburn theory to break it up into a product of matrix rings over division rings, and so the interesting division rings that arise in this way lie a little below the surface.