# Difference between revisions of "Differential ring"

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In [[ring theory]], a '''differential ring''' is a [[ring (mathematics)|ring]] with added structure which generalises the concept of [[derivative]]. | In [[ring theory]], a '''differential ring''' is a [[ring (mathematics)|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]]: | + | Formally, a differential ring is a ring ''R'' with an operation ''D'' on ''R'' which is a [[derivation (mathematics)|derivation]]: |

:<math>D(a+b) = D(a) + D(b) ,\,</math> | :<math>D(a+b) = D(a) + D(b) ,\,</math> |

## Revision as of 22:44, 20 December 2008

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.