PlanetMath: derivation

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derivation

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Let be a commutative ring. A derivation on an -algebra into an -module is an -linear transformation $\d\colon A \to M$ satisfying the properties

$\d (a\mathbf{x}+b\mathbf{y}) = a\,\d\mathbf{x}+ b\,\d\mathbf{y}$
$\d (\mathbf{x}\cdot \mathbf{y}) = \mathbf{x}\cdot \d\mathbf{y}+ \d\mathbf{x}\cdot \mathbf{y}$

for all $a,b \in R$ and $\mathbf{x},\mathbf{y}\in A$ .

"derivation" is owned by djao. [ full author list (2) ]

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Cross-references: properties, transformation, commutative ring
There are 42 references to this entry.

This is version 6 of derivation, born on 2001-12-12, modified 2005-09-15.
Object id is 1089, canonical name is Derivation.
Accessed 7808 times total.

Classification:

AMS MSC:	13N15 (Commutative rings and algebras :: Differential algebra :: Derivations)
	16W25 (Associative rings and algebras :: Rings and algebras with additional structure :: Derivations, actions of Lie algebras)
	17A36 (Nonassociative rings and algebras :: General nonassociative rings :: Automorphisms, derivations, other operators)

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