division ring

A field is a place where you can add, subtract, multiply, and divide by anything nonzero, and the order of multiplication never matters. A division ring is what you get when you keep every one of those abilities — including division — but quietly drop the demand that multiplication commute. So a*b and b*a may differ, yet every nonzero element still has an honest inverse you can divide by. It is a field that has been allowed to be lopsided.

Precisely, a division ring is a ring with identity 1 ≠ 0 in which every nonzero element a has a two-sided multiplicative inverse a^(-1) with a*a^(-1) = a^(-1)*a = 1. Equivalently, its nonzero elements form a group under multiplication. A commutative division ring is exactly a field; the interesting cases are the noncommutative ones, often called skew fields.

These objects are the natural building blocks of the whole subject: the Artin-Wedderburn theorem says every semisimple ring is assembled from matrix rings over division rings, so understanding division rings is half the battle. They behave like fields in many ways — you can still form vector spaces over them and talk about dimension — but you must constantly keep track of sides, since left and right scalars no longer commute past each other.

An honest warning: finiteness kills the noncommutativity. Wedderburn's little theorem proves that every finite division ring is automatically a field. To find a genuinely noncommutative division ring you must go infinite — the smallest classical example, the real quaternions, is four-dimensional over the reals.