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.

Let p be a prime number. A p-derivation or Buium derivative on a ring ${\displaystyle R}$ is a map of sets ${\displaystyle \delta :R\to R}$ 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 ${\displaystyle R}$ has p-torsion.

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