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A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d0312101.png" /> of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d0312102.png" /> into itself which is an endomorphism of the additive group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d0312103.png" /> and satisfies the relation usually referred to as the [[Leibniz rule]]
+
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d0312104.png" /></td> </tr></table>
+
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 +
{{TEX|done}}
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d0312105.png" /> be a left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d0312106.png" />-module. A derivation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d0312107.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d0312108.png" /> is a homomorphism of the respective additive groups which satisfies the condition
+
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]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d0312109.png" /></td> </tr></table>
+
$$
 +
\partial  ( x \cdot y )  = x \partial  ( y) + \partial  ( x) y .
 +
$$
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121010.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121011.png" />. For any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121012.png" /> from the centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121013.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121014.png" />, the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121015.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121016.png" /> is a derivation, is a derivation. The sum of two derivations is also a derivation. This defines the structure of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121017.png" />-module on the set of all derivations in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121018.png" /> with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121019.png" />, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121020.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121021.png" /> is a subring in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121022.png" />, a derivation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121023.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121024.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121025.png" /> is known as an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121027.png" />-derivation. The set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121028.png" />-derivations forms a submodule in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121029.png" />, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121030.png" />. The operation
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121031.png" /></td> </tr></table>
+
$$
 +
\partial  ( x \cdot y )  = x \partial  ( y) + y \partial  ( x)
 +
$$
  
defines the structure of a Lie <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121032.png" />-algebra on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121033.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121034.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121035.png" /> is a homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121036.png" />-modules, then the composition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121037.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121038.png" />.
+
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
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121039.png" /> be a ring of polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121040.png" /> with coefficients in a commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121041.png" />. The mapping
+
$$
 +
[ \partial  , \partial  ^  \prime  ]  = \partial  \circ \partial  ^  \prime  - \partial  ^  \prime  \circ \partial
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121042.png" /></td> </tr></table>
+
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 ) $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121043.png" /></td> </tr></table>
+
Let  $  R $
 +
be a ring of polynomials  $  A [ T _ {1} \dots T _ {n} ] $
 +
with coefficients in a commutative ring  $  A $.  
 +
The mapping
  
is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121044.png" />-derivation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121045.png" />, and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121046.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121047.png" /> is a free module with basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121048.png" />.
+
$$
  
For any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121049.png" /> of an associative ring (or a Lie algebra) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121050.png" /> the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121051.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121052.png" />) is a derivation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121053.png" />, known as an inner derivation. Derivations which are not inner are known as outer.
+
\frac \partial {\partial  T _ {j} }
 +
: F ( T _ {1} \dots T _ {n} ) =
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121054.png" /> is a subring of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121055.png" /> and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121056.png" />, one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121057.png" /> is an extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121058.png" /> if the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121059.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121060.png" /> coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121061.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121062.png" /> is a commutative integral ring and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121063.png" /> is its field of fractions, and also if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121064.png" /> is a separable algebraic extension of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121065.png" /> or if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121066.png" /> is a Lie algebra over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121067.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121068.png" /> is its enveloping algebra, there exists a unique extension of any derivation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121069.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121070.png" />.
+
$$
 +
= \
 +
\sum a _ {i _ {1}  \dots i _ {n} } T _ {1} ^ {i _ {1} } \dots
 +
T _ {n} ^ {i _ {n} }  \rightarrow  \sum a _ {i _ {1}  \dots i _ {n} } i _ {j} T _ {1} ^ {i _ {1} } \dots T _ {j} ^ {i _ {j} - 1 } \dots T _ {n} ^ {i _ {n} }
 +
$$
  
There is a close connection between derivations and ring isomorphisms. Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121071.png" /> is a nilpotent derivation, that is, for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121072.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121073.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121074.png" /> is an algebra over a field of characteristic zero, the mapping
+
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} $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121075.png" /></td> </tr></table>
+
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.
  
is an automorphism of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121076.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121077.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121078.png" /> is a local commutative ring with maximal ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121079.png" />, there is a bijection between the set of derivations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121080.png" /> and the set of automorphisms of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121081.png" /> which induces the identity automorphism of the residue field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121082.png" />. 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]]].
+
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 } $.
  
====References====
+
There is a close connection between derivations and ring isomorphisms. Thus, if  $  \partial  $
<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,   "Strucuture of arbitrary purely inseparable extension fields" , Springer  (1970)</TD></TR></table>
+
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====
 
====Comments====
The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121083.png" />-derivations in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121084.png" /> are precisely the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121085.png" />-linear mappings from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121086.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121087.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121088.png" />-algebra, then a derivation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121089.png" /> is a [[Crossed homomorphism|crossed homomorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121090.png" /> or, equivalently, a Hochschild <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121092.png" />-cocycle.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121093.png" /> is semi-simple, all derivations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121094.png" /> are inner, i.e. in that case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121095.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121096.png" /> be any algebra (or ring), not necessarily commutative or associative. The algebra is said to be Lie admissible if the associated algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121097.png" /> with multiplication <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121098.png" /> 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.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d03121099.png" /> together with a derivation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031210/d031210100.png" /> is a [[Differential ring|differential ring]], cf. also [[Differential-algebra(2)|Differential algebra]] and [[Differential field|Differential field]].
+
A ring $  R $
 +
together with a derivation $  \partial  $
 +
is a [[differential ring]], cf. also [[Differential-algebra(2)|Differential algebra]] and [[Differential field]].

Latest revision as of 17:32, 5 June 2020


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.

How to Cite This Entry:
Derivation in a ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Derivation_in_a_ring&oldid=26603
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article