Difference between revisions of "Differential ring"

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==Examples==
 
==Examples==
 
* Every ring is a differential ring with the zero map as derivation.
 
* Every ring is a differential ring with the zero map as derivation.
* The [[formal derivative]] makes the polynomial ring ''R''[''X'']] over ''R'' a differential ring with
+
* The [[formal derivative]] makes the polynomial ring ''R''[''X''] over ''R'' a differential ring with
 
:<math>D(X^n) = n.X^{n-1} ;\,</math>
 
:<math>D(X^n) = n.X^{n-1} ;\,</math>
 
:<math>D(r) = 0 \mbox{ for } r \in R.\,</math>
 
:<math>D(r) = 0 \mbox{ for } r \in R.\,</math>

Revision as of 22:41, 20 December 2008

In ring theory, a differential ring is a ring with added structure which generalises the concept of derivative.

Formally, a differential ring is a ring R with an operation D on R which is a derivation:

Examples

  • Every ring is a differential ring with the zero map as derivation.
  • The formal derivative makes the polynomial ring R[X] over R a differential ring with

Ideals

A differential ring homomorphism is a ring homomorphism f from differential ring (R,D) to (S,d) such that f.D = d.f. A differential ideal is an ideal I of R such that D(I) is contained in I.

References