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 p-derivation is related to that of a derivation in differential algebra.
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
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.
- 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.
- Buium, Alex (1989), Arithmetic Differential Equations, Mathematical Surveys and Monographs, Springer-Verlag, ISBN 0-8218-3862-8.