$p$ -powers of derivations

PROPOSITION 6.1   Let $p$ be a prime number. Let $A$ be a (not necessarily commutative, but unital, associative) algebra over $\mathbb{F}_p$ . Let $B$ be a (not necessarily commutative, but unital, associative) algebra over $A$ . Then for any $A$ -derivation $D: B\to B$ , its $p$ -power $D^p$ is also an derivation.

PROOF.. That $D^p$ is $A$ -linear is clear. Let $f,g \in B$ . Then for any positive integer $k$ , we have by using the Leibniz rule of $D$ ,

$$D^k (f g)=\sum_{j=0}^k \binom{k}{j} D^j (f) D^{k-j}(g).$$

In particular, when $k=p$ , this means

$$D^p (f g)=D^p (f) g + f D^p(g).$$

Thus $D^p$ also satisfies the Leibniz rule. $\qedsymbol$