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

$D(X^{n})=n.X^{n-1};\,$

$D(r)=0{\mbox{ for }}r\in R.\,$

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.