Noun
central simple algebra (plural central simple algebras)
(algebra, ring theory) A finite-dimensional associative algebra over some field K that is a simple algebra and whose centre is exactly K.
The complex numbers
C
{\displaystyle \mathbb {C} }
form a central simple algebra over themselves, but not over the real numbers
R
{\displaystyle \mathbb {R} }
(the centre of
C
{\displaystyle \mathbb {C} }
is all of
C
{\displaystyle \mathbb {C} }
, not just
R
{\displaystyle \mathbb {R} }
). The quaternions
H
{\displaystyle \mathbb {H} }
form a 4-dimensional central simple algebra over
R
{\displaystyle \mathbb {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).