Central simple algebra

A central simple algebra over a field $F$ is a nonzero finite-dimensional associative unital $F$ -algebra $A$ such that its center is $F1_A$ and $A$ is simple: its only two-sided ideals are $0$ and $A$ .

Both the base field and the algebra structure are part of this assertion.

Examples

The matrix algebra $M_n(F)$ , $n\ge1$ , is central simple. A nonzero two-sided ideal contains every matrix unit after left and right multiplication by matrix units and scalar rescaling; a matrix commuting with all matrix units must be scalar.

Every quaternion algebra over $F$ is central simple, including both division and split examples. Thus “simple” does not imply that every nonzero element is invertible.

A nontrivial field extension $K/F$ , viewed as an $F$ -algebra, is simple but not central over $F$ , since its center is $K$ .

Change of scalars

Central simplicity persists under extension of the base field. Over an algebraic closure, a central simple algebra becomes a matrix algebra. Brauer classes record its structure up to matrix-algebra factors.

References

John Voight, Quaternion Algebras, author's text, §§7.2 and 7.5–7.6, especially Lemma 7.5.4 and Proposition 7.6.1.