Logarithmable automorphism

Definition
Suppose $$R$$ is a non-associative ring and $$\alpha$$ is an automorphism of $$R$$. Suppose further that:


 * 1) $$\alpha -1$$ is locally nilpotent, i.e., for every $$x \in R$$, there exists a natural number $$n$$, possibly dependent upon $$x$$, such that $$(\alpha - 1)^n(x) = 0$$.
 * 2) $$\alpha - 1$$ is an infinitely powered endomorphism of the additive structure of $$R$$, i.e., for all natural numbers $$n$$, $$(\alpha - 1)^n(R)$$ is powered for all primes less than or equal to $$n$$.
 * 3) The logarithm of $$\alpha$$, which can be defined because of (1) and (2), is a derivation of $$R$$.

We then say that $$\alpha$$ is a logarithmabl automorphism of $$R$$.