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]].   
  
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:<math>D(a+b) = D(a) + D(b) ,\,</math>
 
:<math>D(a+b) = D(a) + D(b) ,\,</math>
:<math>D(a.b) = D(a).b + a.D(b) . \,</math>
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:<math>D(a \cdot b) = D(a) \cdot b + a \cdot D(b) . \,</math>
  
 
==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>
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::<math>D(X^n) = nX^{n-1} ,\,</math>
:<math>D(r) = 0 \mbox{ for } r \in R.\,</math>
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::<math>D(r) = 0 \mbox{ for } r \in R.\,</math>
  
==Ideals==
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==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''.
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A ''differential ring homomorphism'' is a ring homomorphism ''f'' from differential ring (''R'',''D'') to (''S'',''d'') such that ''f''&middot;''D'' = ''d''&middot;''f''.  A ''differential ideal'' is an ideal ''I'' of ''R'' such that ''D''(''I'') is contained in ''I''.
 
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==References==
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* {{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 }}
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* {{cite book | author=Bruno Poizat | title=Model Theory | publisher=[[Springer Verlag]] | year=2000 | isbn=0-387-98655-3 | pages=71 }}
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Latest revision as of 16:31, 12 June 2009

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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.