Logarithm of automorphism is derivation under suitable nilpotency assumptions

Weaker version: global powering and torsion assumptions
Suppose $$R$$ is a non-associative ring and $$\alpha$$ is an automorphism of $$R$$. Suppose the following hold:


 * 1) $$\alpha - 1$$ is nilpotent.
 * 2) The nilpotency of $$\alpha - 1$$ is at most one more than the powering threshold for $$R$$. In other words, there exists a natural number $$n$$ such that $$(\alpha - 1)^n = 0$$ and $$R$$ is powered for all primes strictly less than $$n$$.
 * 3) The binilpotency of $$\alpha - 1$$ is at most one more than the torsion-free threshold for $$R$$. In other words, there exists a natural number $$m$$ such that $$(\alpha - 1)^i(x)(\alpha - 1)^j(y) = 0$$ for all $$x,y \in R, i + j \ge m$$, and $$R$$ is $$p$$-torsion-free for all primes $$p < m$$.

Then, we claim that $$\alpha$$ is a logarithmable automorphism. Explicitly, we are claiming that $$\log \alpha$$ is a derivation of $$R$$, where:

$$\log \alpha := (\alpha - 1) - \frac{(\alpha - 1)^2}{2} + \frac{(\alpha - 1)^3}{3} - \dots + (-1)^n\frac{(\alpha - 1)^{n-1}}{n - 1}$$

Stronger version: image powering and torsion assumptions
Suppose $$R$$ is a non-associative ring (i.e., a not necessarily associative ring) and $$\alpha$$ is an automorphism of $$R$$. Suppose the following hold:


 * 1) $$\alpha -1$$ is locally nilpotent.
 * 2) $$\alpha - 1$$ is an infinitely powered endomorphism of $$R$$, i.e., the  powering threshold for $$\alpha - 1$$ is $$\infty$$.
 * 3) $$\alpha -1$$ is an infinitely bi-torsion-free endomorphism of $$R$$, i.e., the  bi-torsion-free threshold for $$d$$ is $$\infty$$.

Then, we claim that $$\alpha$$ is a logarithmable automorphism. Explicitly, we are claiming that the logarithm of $$\alpha$$ exists and is a derivation.

Similar facts

 * Exponential of derivation is automorphism under suitable nilpotency assumptions