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 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 $i+j\geq n$.

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 $\implies$ $\theta ^{n}(R)*R=R*\theta ^{n}(R)=0$