Central Simple Algebra Definition

The complex numbers \C form a central simple algebra over themselves, but not over the real numbers \R (the centre of \C is all of \C, not just \R). The quaternions \mathbb H form a 4-dimensional central simple algebra over \R.

The concept of central simple algebra over a field K represents a noncommutative analogue to that of extension field over K. In both cases, the object has no nontrivial two-sided ideals and has a distinguished field in its centre, although a central simple algebra need not be commutative and need not have inverses (does not have be a division algebra ).