# Derivation in a ring

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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)

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.