Differential ring

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:


 * $$D(a+b) = D(a) + D(b) ,\,$$
 * $$D(a.b) = D(a).b + a.D(b) . \,$$

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

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.