Infinitely bi-torsion-free additive endomorphism of a non-associative ring

Definition

Suppose $R$ is a non-associative ring and $f:R\to R$ is an endomorphism of the additive group of $R$. We say that $f$ is infinitely bi-torsion-free if, for all natural numbers $n$, and all $i,j$ such that $i+j=n$, the torsion-free threshold for the subring of $R$ generated by $f^{i}(R)*f^{j}(R)$ is at least equal to $n$.

This is equivalent to saying that the bi-torsion-free threshold for $f$ is $\infty$.