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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\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$.



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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 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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