Difference between revisions of "Differential ring"

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In [[ring theory]], a '''differential ring''' is a [[ring (mathematics)|ring]] with added structure which generalises the concept of [[derivative]].   
 
In [[ring theory]], a '''differential ring''' is a [[ring (mathematics)|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]]:
+
Formally, a differential ring is a ring ''R'' with an operation ''D'' on ''R'' which is a [[derivation (mathematics)|derivation]]:
  
 
:<math>D(a+b) = D(a) + D(b) ,\,</math>
 
:<math>D(a+b) = D(a) + D(b) ,\,</math>

Revision as of 22:44, 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