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
R
is a map of sets
δ
:
R
→
R
that satisfies the following "product rule":

δ
p
(
a
b
)
=
δ
p
(
a
)
b
p
+
a
p
δ
p
(
b
)
+
p
δ
p
(
a
)
δ
p
(
b
)
and "sum rule":

δ
p
(
a
+
b
)
=
δ
p
(
a
)
+
δ
p
(
b
)
+
a
p
+
b
p
−
(
a
+
b
)
p
p
.

as well as

δ
p
(
1
)
=
0
.

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
R
has *p*-torsion.

A map
σ
:
R
→
R
is a lift of the Frobenius endomorphism provided
σ
(
x
)
=
x
p
mod
p
R
. An example such lift could come from the Artin map.

If
(
R
,
δ
)
is a ring with a *p*-derivation, then the map
σ
(
x
)
:=
x
p
+
p
δ
(
x
)
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
R
=
Z
the unique *p*-derivation is the map
δ
(
x
)
=
x
−
x
p
p
.
The quotient is well-defined because of Fermat's Little Theorem.

If *R* is any *p*-torsion free ring and
σ
:
R
→
R
is a lift of the Frobenius endomorphism then
δ
(
x
)
=
σ
(
x
)
−
x
p
p
defines a *p*-derivation.