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$ 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$ ,

$\displaystyle 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

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

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



2012-02-29