**Derivation (abstract Algebra)**

In abstract algebra, a **derivation** is a function on an algebra which generalizes certain features of the derivative operator. Specifically, given an algebra *A* over a ring or a field *K*, a *K*-derivation is a *K*-linear map *D*: *A* → *A* that satisfies Leibniz's law:

More generally, a *K*-linear map *D* of *A* into an *A*-module *M*, satisfying the Leibniz law is also called a derivation. The collection of all *K*-derivations of *A* to itself is denoted by Der_{K}(*A*). The collection of *K*-derivations of *A* into an *A*-module *M* is denoted by Der_{K}(*A*,*M*).

Derivations occur in many different contexts in diverse areas of mathematics. The partial derivative with respect to a variable is an **R**-derivation on the algebra of real-valued differentiable functions on **R**n. The Lie derivative with respect to a vector field is an **R**-derivation on the algebra of differentiable functions on a differentiable manifold; more generally it is a derivation on the tensor algebra of a manifold. The Pincherle derivative is an example of a derivation in abstract algebra. If the algebra *A* is noncommutative, then the commutator with respect to an element of the algebra *A* defines a linear endomorphism of *A* to itself, which is a derivation over *K*. An algebra *A* equipped with a distinguished derivation *d* forms a differential algebra, and is itself a significant object of study in areas such as differential Galois theory.

Read more about Derivation (abstract Algebra): Properties, Graded Derivations

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