Derivation in a ring

A mapping of a ring into itself which is an endomorphism of the additive group of and satisfies the relation usually referred to as the Leibniz rule

Let be a left -module. A derivation in with values in is a homomorphism of the respective additive groups which satisfies the condition

for all from . For any element from the centre of , the mapping , where is a derivation, is a derivation. The sum of two derivations is also a derivation. This defines the structure of a -module on the set of all derivations in with values in , denoted by . If is a subring in , a derivation such that for all is known as an -derivation. The set of all -derivations forms a submodule in , denoted by . The operation

defines the structure of a Lie -algebra on the -module . If is a homomorphism of -modules, then the composition for any .

Let be a ring of polynomials with coefficients in a commutative ring . The mapping

is an -derivation in , and the -module is a free module with basis .

For any element of an associative ring (or a Lie algebra) the mapping (or ) is a derivation in , known as an inner derivation. Derivations which are not inner are known as outer.

If is a subring of a ring and if , one says that is an extension of if the restriction of to coincides with . If is a commutative integral ring and is its field of fractions, and also if is a separable algebraic extension of the field or if is a Lie algebra over a field and is its enveloping algebra, there exists a unique extension of any derivation to .

There is a close connection between derivations and ring isomorphisms. Thus, if is a nilpotent derivation, that is, for some , , and is an algebra over a field of characteristic zero, the mapping

is an automorphism of the -algebra . If is a local commutative ring with maximal ideal , there is a bijection between the set of derivations and the set of automorphisms of the ring which induces the identity automorphism of the residue field . Derivations of non-separable field extensions play the role of elements of the Galois group of separable extensions in the Galois theory of such extensions [4].

References

[1]	N. Bourbaki, "Algebra" , Elements of mathematics , 1 , Addison-Wesley (1973) (Translated from French)
[2]	N. Jacobson, "The theory of rings" , Amer. Math. Soc. (1943)
[3]	S. Lang, "Algebra" , Addison-Wesley (1974)
[4]	J. Mordeson, B. Vinograde, "Structure of arbitrary purely inseparable extension fields" , Springer (1970)

Comments

The -derivations in are precisely the -linear mappings from . If is an -algebra, then a derivation in is a crossed homomorphism or, equivalently, a Hochschild -cocycle.

If the Lie algebra is semi-simple, all derivations are inner, i.e. in that case .

Let be any algebra (or ring), not necessarily commutative or associative. The algebra is said to be Lie admissible if the associated algebra with multiplication is a Lie algebra. Associative algebras and Lie algebras are Lie admissible, but there are also other examples. These algebras were introduced by A.A. Albert in 1948.

A ring together with a derivation is a differential ring, cf. also Differential algebra and Differential field.