From Wolfram Alpha I have that the definition of a division ring is;
"a ring in which every nonzero element has a multiplicative inverse, but multiplication is not necessarily commutative". Can someone give me an example of a division ring where multiplication is commutative and one where it is not commutative?
A division ring ring with commutative multiplication is the same thing as a field, so examples include $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$. (In fact, division rings were once called fields, and fields, commutative division rings; division rings are sometimes called skew fields.)
The standard example of a division ring is the quaternions $\mathbb{H}$. A countable example is the set of quaternions with rational coefficients. By Wedderburn's (Little) Theorem, any finite division ring is necessarily commutative.
Edit Wikipedia has another example built using formal Laurent series and complex conjugation, and Jyrki Lahtonen gave a more exotic example here: An example of a division ring $D$ that is **not** isomorphic to its opposite ring .
