Differential ring: Difference between revisions
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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 | :<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) = | ::<math>D(X^n) = nX^{n-1} ,\,</math> | ||
:<math>D(r) = 0 \mbox{ for } r \in R.\,</math> | ::<math>D(r) = 0 \mbox{ for } r \in R.\,</math> | ||
== | ==Ideal== | ||
A ''differential ring homomorphism'' is a ring homomorphism ''f'' from differential ring (''R'',''D'') to (''S'',''d'') such that ''f'' | 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''.[[Category:Suggestion Bot Tag]] | ||
Latest revision as of 06:01, 7 August 2024
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.