Algebra over a field

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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.