Derivation (mathematics)

In mathematics, a derivation is a map which has formal algebraic properties generalising those of the derivative.

Let R be a ring (mathematics) and A an R-algebra (A is a ring containing a copy of R in the centre). A derivation is an R-linear map D with the property that

$D(ab)=a.D(b)+D(a).b.\,$

The constants of D are the elements mapped to zero. The constants include the copy of R inside A.

A derivation "on" A is a derivation from A to A.

Linear combinations of derivations are again derivations, so the derivations from A to M form an R-module, denoted Der_R(A,M).

Examples

• The zero map is a derivation.
• The formal derivative is a derivation on the polynomial ring R[X] with constants R.

Universal derivation

There is a universal derivation Ω such that

$\operatorname {Der} _{R}(A,M)=\operatorname {Hom} _{A}(\Omega ,M)\,$

as a functorial isomorphism.

References

• Serge Lang (1993). Algebra, 3rd ed. Addison-Wesley, 746-749. ISBN 0-201-55540-9.