Differential ring

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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:

D(a+b) = D(a) + D(b) ,\,
D(a \cdot b) = D(a) \cdot b + a \cdot D(b) . \,

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
D(X^n) = nX^{n-1} ,\,
D(r) = 0 \mbox{ for } r \in R.\,

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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