# Difference between revisions of "Differential ring"

From Citizendium

(new entry, just a stub) |
Jitse Niesen (Talk | contribs) (use \cdot for multiplication) |
||

(4 intermediate revisions by one other user not shown) | |||

Line 1: | Line 1: | ||

+ | {{subpages}} | ||

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> | ||

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

==Examples== | ==Examples== | ||

* Every ring is a differential ring with the zero map as derivation. | * Every ring is a differential ring with the zero map as derivation. | ||

− | * The [[formal derivative]] makes the polynomial ring ''R''[''X'' | + | * The [[formal derivative]] makes the polynomial ring ''R''[''X''] over ''R'' a differential ring with |

− | :<math>D(X^n) = | + | ::<math>D(X^n) = nX^{n-1} ,\,</math> |

− | :<math>D(r) = 0 \mbox{ for } r \in R.\,</math> | + | ::<math>D(r) = 0 \mbox{ for } r \in R.\,</math> |

− | == | + | ==Ideal== |

− | A ''differential ring homomorphism'' is a ring homomorphism ''f'' from differential ring (''R'',''D'') to (''S'',''d'') such that ''f'' | + | 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''. |

− | + | ||

− | + | ||

− | + | ||

− | + |

## Latest revision as of 16:31, 12 June 2009

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

## 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*.