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 p-derivation is related to that of a derivation in differential algebra. Definition 
Let p be a prime number. A p-derivation or Buium derivative on a ring  is a map  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 of such a 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  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. See also 
 Witt vector
 Arithmetic derivative
 Derivation
 Fermat quotient

References 
 . External links 
 Project Euclid 

Differential algebra
Generalizations of the derivative