Binilpotency

Definition
Suppose $$R$$ is a non-associative ring and $$\theta: R \to R$$ is an endomorphism of the additive group of $$R$$. Suppose $$n$$ is a nonnegative integer. We say that $$\theta$$ is $$n$$-step-binilpotent if the following equivalent conditions hold (note that we define $$\theta^0$$ as the identity map):


 * $$\theta^i(x) * \theta^j(y) = 0$$ for all $$x,y \in R$$ and all nonnegative integers $$i,j$$ with $$i + j = n$$.
 * $$\theta^i(x) * \theta^j(y) = 0$$ for all $$x,y \in R$$ and all nonnegative integers $$i,j$$ with $$i + j \ge n$$.

The binilpotency of $$\theta$$ is defined as the smallest nonnegative integer $$n$$ for which $$\theta$$ is $$n$$-step-binilpotent.

See also weak binilpotency.

Relation with nilpotency
We have:

$$n$$-step-nilpotent $$\implies $$ $$(2n - 1)$$-step-binilpotent

Conversely:

$$n$$-step-binilpotent $$\implies $$ $$\theta^n(R) * R = R * \theta^n(R) = 0$$