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derivation (Definition)

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

  • $\d(a\x+b\y) = a\,\d\x + b\,\d\y$
  • $\d(\x\cdot \y) = \x \cdot \d\y + \d\x \cdot \y$
for all $a,b \in R$ and $\x,\y \in A$ .




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derivations on a ring of continous functions (Example) by joking
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Cross-references: properties, transformation, commutative ring
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This is version 6 of derivation, born on 2001-12-12, modified 2005-09-15.
Object id is 1089, canonical name is Derivation.
Accessed 10154 times total.

Classification:
AMS MSC13N15 (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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