# 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:

${\displaystyle D(a+b)=D(a)+D(b),\,}$
${\displaystyle D(a\cdot b)=D(a)\cdot b+a\cdot D(b).\,}$

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

## 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.