# Differential ring

From Citizendium

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

## 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*.

## References

- Andy R. Magid (1994).
*Lectures on Differential Galois Theory*. AMS Bookstore, 1-2. ISBN 0-8218-7004-1. - Bruno Poizat (2000).
*Model Theory*. Springer Verlag, 71. ISBN 0-387-98655-3.