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
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Let be a left
-module. A derivation in
with values in
is a homomorphism of the respective additive groups which satisfies the condition
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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
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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
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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
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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, "Strucuture 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.
Derivation in a ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Derivation_in_a_ring&oldid=26603