Nilpotent element with divided powers

Definition
Suppose $$R$$ is an associative ring. A nilpotent element with divided powers is an defining ingredient::element with divided powers $$\mathbf{x} = (x^{(1)},x^{(2)},\dots)$$ such that there exists a positive integer $$n$$ such that $$x^{(m)} = 0$$ for all $$m \ge n$$. The smallest $$n$$ that works is termed the nilpotency of $$\mathbf{x}$$.

Facts

 * Suppose the additive group of $$R$$ is torsion-free. Then, $$\mathbf{x} = (x^{(1)},x^{(2)},\dots)$$ is a nilpotent element with divided powers if and only if $$x^{(1)}$$ is a nilpotent element of $$R$$. Moreover, they have the same nilpotency.
 * Suppose $$x \in R$$ is an element with nilpotency $$n$$, i.e., $$x^n = 0$$ and no smaller power of $$x$$ is $$0$$. Suppose the powering threshold for $$R$$ is at least $$n - 1$$. Then, we can define a nilpotent element with divided powers $$(x,x^2/2!,\dots,x^{n-1}/(n-1)!,0,0,\dots)$$.