# Derivation (mathematics)

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

Let *R* be a ring 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* from *A* to some *A*-module *M* with the property that

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,*d*) with a universal property. Given a derivation *D*:*A* → *M*, there is a unique *A*-linear *f*:Ω → *M* such that *D* = *d*·*f*. Hence

as a functorial isomorphism.

Consider the multiplication map μ on the tensor product (over *R*)

defined by . Let *J* be the kernel of μ. We define the module of *differentials*

as an ideal in , where the *A*-module structure is given by *A* acting on the first factor, that is, as . We define the map *d* from *A* to Ω by

- .

This is the universal derivation.

## Kähler differentials

A Kähler differential, or formal differential form, is an element of the universal derivation space Ω, hence of the form Σ_{i} *x _{i}*

*dy*. An

_{i}*exact*differential is of the form for some

*y*in

*A*. The exact differentials form a submodule of Ω.