Definition 59.6.1. Let $A$ be a ring. Let $(B, J, \delta )$ be a divided power ring. Let $A \to B$ be a ring map. Let $M$ be an $B$-module. A divided power $A$-derivation into $M$ is a map $\theta : B \to M$ which is additive, annihilates the elements of $A$, satisfies the Leibniz rule $\theta (bb') = b\theta (b') + b'\theta (b)$ and satisfies
for all $n \geq 1$ and all $x \in J$.