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.
A p-derivation or Buium derivative on a ring
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
is a lift of the Frobenius endomorphism provided
is a ring with a p-derivation, then the map
When the ring R is p-torsion free the correspondence is a bijection.
The quotient is well-defined because of Fermat's little theorem.