Weak binilpotency

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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 positive integer. We say that $\theta$ is $n$-step-weak binilpotent if the following holds:

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

Note that if $n>1$, then this is equivalent to checking that:

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

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