linear algebra over a division ring vs. over a field 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...

Let me quote a relevant paragraph in the Wikipedia article on "Division ring":

Much of linear algebra may be
  formulated, and remains correct, for
  (left) modules over division rings
  instead of vector spaces over fields.
  Every module over a division ring has
  a basis; linear maps between
  finite-dimensional modules over a
  division ring can be described by
  matrices, and the Gaussian elimination
  algorithm remains applicable.
  Differences between linear algebra
  over fields and skew fields occur
  whenever the order of the factors in a
  product matters. For example, the
  proof that the column rank of a matrix
  over a field equals its row rank
  yields for matrices over division
  rings only that the left column rank
  equals its right row rank: it does not
  make sense to speak about the rank of
  a matrix over a division ring.

I hope this helps!
