#
*p*-derivation

In mathematics, more specifically differential algebra, a ** p-derivation** (for

*p*a prime number) on a ring

*R*, is a mapping from

*R*to

*R*that satisfies certain conditions outlined directly below. The notion of a

**is related to that of a derivation in differential algebra.**

*p*-derivation## Definition

Let *p* be a prime number. A ** p-derivation** or Buium derivative on a ring is a map of sets that satisfies the following "product rule":

and "sum rule":

- .

as well as

- .

Note that in the "sum rule" we are not really dividing by *p*, since all the relevant binomial coefficients in the numerator are divisible by *p*, so this definition applies in the case when has *p*-torsion.

## Relation to Frobenius Endomorphisms

A map is a lift of the Frobenius endomorphism provided . An example such lift could come from the Artin map.

If is a ring with a *p*-derivation, then the map
defines a ring endomorphism which is a lift of the Frobenius endomorphism. When the ring *R* is *p*-torsion free the correspondence is a bijection.

## Examples

- For the unique
*p*-derivation is the map

The quotient is well-defined because of Fermat's Little Theorem.

- If
*R*is any*p*-torsion free ring and is a lift of the Frobenius endomorphism then

defines a *p*-derivation.

## References

- Buium, Alex (1989),
*Arithmetic Differential Equations*, Mathematical Surveys and Monographs, Springer-Verlag, ISBN 0-8218-3862-8.