Derivation with divided powers

Usual notion: derivation with divided powers for all natural numbers
Suppose $$R$$ is a non-associative ring (i.e., a not necessarily associative ring). A derivation with divided powers for $$R$$ is the following data:

satisfying the following compatibility conditions:

where the multiplication on both sides is multiplication in the ring $$R$$.

The derivation in question is $$d^{(1)} = d$$, and this is indeed a derivation by the second condition applied to $$n = 1$$.

Intuitively, $$d^{(n)}$$ is trying to be $$d^n/n!$$. Indeed, if $$R$$ is a $$\mathbb{Q}$$-algebra, then we can set $$d^{(n)} = d^n/n!$$ for all $$n$$, and this is the only feasible choice.

Derivation with divided powers up to a point
Suppose $$R$$ is a non-associative ring (i.e., a not necessarily associative ring) and $$m$$ is a natural number. A derivation with divided powers up to $$m$$ (which we may call a derivation with partial divided powers) for $$R$$ is the following data:

satisfying the following compatibility conditions:

where the multiplication on both sides is multiplication in the ring $$R$$.

The derivation in question is $$d^{(1)} = d$$, and this is indeed a derivation by the second condition applied to $$n = 1$$.

Facts

 * Exponential of nilpotent derivation with divided powers is automorphism

Related notions

 * Derivation with divided Leibniz condition powers: Sequence satisfying the Leibniz rule conditions but not necessarily the divided power conditions.