Global LUCS-Lazard Lie ring

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This article defines a Lie ring property: a property that can be evaluated to true/false for any Lie ring.
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Definition

A Lie ring is termed a global LUCS-Lazard Lie ring if there exists a positive integer $c$ such that:

1. $G$ is a nilpotent Lie ring of nilpotency class at most $c$.
2. The following equivalent formulations:
   • For every positive integer $i$, every element of the $i^{th}$ lower central series member ${\gamma }_i(L)$ has a unique $p^{th}$ root in $L$ for all $p\le i$.
   • For every positive integer $i$, every element of the $i^{th}$ lower central series member ${\gamma }_i(L)$ has a unique $p^{th}$ root in the upper central series member $Z^{c+1-i}(L)$ for all $p\le i$.
   • For every positive integer $i$, every element of the $i^{th}$ lower central series member ${\gamma }_i(L)$ has unique $p^{th}$ roots in $G$ for all $p\le i$, and this root is inside $Z^{c+1-i}(L)$.