Difference between revisions of "Differential ring"

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==Ideal==
 
==Ideal==
 
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''.
 
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==
 
* {{cite book | title=Lectures on Differential Galois Theory | author=Andy R. Magid | publisher=AMS Bookstore | year=1994 | isbn=0-8218-7004-1 | pages=1-2 }}
 
* {{cite book | author=Bruno Poizat | title=Model Theory | publisher=[[Springer Verlag]] | year=2000 | isbn=0-387-98655-3 | pages=71 }}
 

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This editable Main Article is under development and subject to a disclaimer.

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

Ideal

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.