# Algebra over a field

From Citizendium

In abstract algebra, an **algebra over a field** *F*, or *F*-**algebra** is a ring *A* containing an isomorphic copy of *F* in the centre. Another way of expressing this is to say that *A* is a vector space over *F* equipped with a further algebraic structure of multiplication compatible with the vector space structure.

## Examples

- Any extension field
*E*/*F*can be regarded as an*F*-algebra. - The matrix ring
*M*_{n}(*F*) of*n*×*n*square matrices with entries in*F*is an*F*-algebra, with*F*embedded as the scalar matrices. - The ring of quaternions
**H**is a division ring with centre the real numbers**R**. It may thus be regarded as an**R**-algebra.

## References

- Serge Lang (1993).
*Algebra*, 3rd ed. Addison-Wesley. ISBN 0-201-55540-9.