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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 [[#References|[4]]].

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

====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|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 algebra|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(2)|Differential algebra]] and [[Differential field]].