Algebra over a field

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 quaternion ring is an algebra over the field of real numbers.

References

• Serge Lang (1986). Algebraic number theory. Springer-Verlag. ISBN 0-387-94225-4.