Theorem 2

— Let $A$ be a commutative ring of characteristic $p$. The mapping $a \mapsto a^p$ is an endomorphism of the ring $A$, that is, we have the relations

(1)

(2)

for $a, b$ in $A$.

Formula (2) follows from the commutativity of $A$. To prove (1) we use the binomial formula $(a + b)^p = a^p + b^p + \sum_{i=1}^{p-1} \binom{p}{i} \cdot a^i b^{p-i}$; since $p \cdot x = 0$ for all $x \in A$, it suffices to prove the following lemma: