Derivation in a ring

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

$$ \partial ( x \cdot y ) = x \partial ( y) + \partial ( x) y . $$

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

$$ \partial ( x \cdot y ) = x \partial ( y) + y \partial ( x) $$

for all $ x , y $ from $ R $. For any element $ c $ from the centre $ C $ of $ R $, the mapping $ x \rightarrow c \partial ( x) $, where $ \partial $ is a derivation, is a derivation. The sum of two derivations is also a derivation. This defines the structure of a $ C $- module on the set of all derivations in $ R $ with values in $ M $, denoted by $ \mathop{\rm Der} ( R , M ) $. If $ S $ is a subring in $ R $, a derivation $ \partial $ such that $ \partial ( s) = 0 $ for all $ s \in S $ is known as an $ S $- derivation. The set of all $ S $- derivations forms a submodule in $ \mathop{\rm Der} ( R , M ) $, denoted by $ \mathop{\rm Der} _ {S} ( R , M ) $. The operation

$$ [ \partial , \partial ^ \prime ] = \partial \circ \partial ^ \prime - \partial ^ \prime \circ \partial $$

defines the structure of a Lie $ S $- algebra on the $ S $- module $ \mathop{\rm Der} _ {S} ( R , M ) $. If $ \phi : R \rightarrow M $ is a homomorphism of $ R $- modules, then the composition $ \phi \circ \partial \in \mathop{\rm Der} ( R , M ) $ for any $ \partial \in \mathop{\rm Der} ( R , R ) $.

Let $ R $ be a ring of polynomials $ A [ T _ {1} \dots T _ {n} ] $ with coefficients in a commutative ring $ A $. The mapping

$$ \frac \partial {\partial T _ {j} } : F ( T _ {1} \dots T _ {n} ) = $$

$$ = \ \sum a _ {i _ {1} \dots i _ {n} } T _ {1} ^ {i _ {1} } \dots T _ {n} ^ {i _ {n} } \rightarrow \sum a _ {i _ {1} \dots i _ {n} } i _ {j} T _ {1} ^ {i _ {1} } \dots T _ {j} ^ {i _ {j} - 1 } \dots T _ {n} ^ {i _ {n} } $$

is an $ A $- derivation in $ R $, and the $ R $- module $ \mathop{\rm Der} _ {A} ( R , R ) $ is a free module with basis $ \partial / \partial T _ {1} \dots \partial / \partial T _ {n} $.

For any element $ a $ of an associative ring (or a Lie algebra) $ R $ the mapping $ x \rightarrow ax - xa $( or $ x \rightarrow ax $) is a derivation in $ R $, known as an inner derivation. Derivations which are not inner are known as outer.

If $ R $ is a subring of a ring $ R ^ { \prime } $ and if $ \partial \in \mathop{\rm Der} ( R , R ) $, one says that $ \overline \partial \; \in \mathop{\rm Der} ( R ^ { \prime } , R ^ { \prime } ) $ is an extension of $ \partial $ if the restriction of $ \overline \partial \; $ to $ R $ coincides with $ \partial $. If $ R $ is a commutative integral ring and $ R ^ { \prime } $ is its field of fractions, and also if $ R ^ { \prime } $ is a separable algebraic extension of the field $ R $ or if $ R $ is a Lie algebra over a field $ k $ and $ R ^ { \prime } $ is its enveloping algebra, there exists a unique extension of any derivation $ \partial : R \rightarrow R $ to $ R ^ { \prime } $.

There is a close connection between derivations and ring isomorphisms. Thus, if $ \partial $ is a nilpotent derivation, that is, for some $ n $, $ \partial ^ {n} = 0 $, and $ R $ is an algebra over a field of characteristic zero, the mapping

$$ \mathop{\rm exp} ( \partial ) = 1 + \partial + \frac{\partial ^ {2} }{2!} + \dots + \frac{\partial ^ {n-} 1 }{( n - 1 ) ! } $$

is an automorphism of the $ k $- algebra $ R $. If $ R $ is a local commutative ring with maximal ideal $ \mathfrak m $, there is a bijection between the set of derivations $ \mathop{\rm Der} ( R , R / \mathfrak m ) $ and the set of automorphisms of the ring $ R / \mathfrak m ^ {2} $ which induces the identity automorphism of the residue field $ R / \mathfrak m $. 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 $ S $- derivations in $ \mathop{\rm Der} _ {S} ( R , M ) $ are precisely the $ S $- linear mappings from $ \mathop{\rm Der} ( R , M ) $. If $ A $ is an $ R $- algebra, then a derivation in $ \mathop{\rm Der} ( A , R ) $ is a crossed homomorphism $ A \rightarrow R $ or, equivalently, a Hochschild $ 1 $- cocycle.

If the Lie algebra $ \mathfrak g $ is semi-simple, all derivations $ \mathfrak g \rightarrow \mathfrak g $ are inner, i.e. in that case $ \mathop{\rm Der} ( \mathfrak g , \mathfrak g ) \simeq \mathfrak g $.

Let $ R $ be any algebra (or ring), not necessarily commutative or associative. The algebra is said to be Lie admissible if the associated algebra $ \overline{R}\; $ with multiplication $ [ a , b ] = a b - b a $ 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 $ R $ together with a derivation $ \partial $ is a differential ring, cf. also Differential algebra and Differential field.