Lazard Lie ring

Definition
A Lazard Lie ring is a Lie ring $$L$$ such that there exists a natural number $$c$$ with both the following two properties:

Condition (1) gets more demanding (i.e., stronger, so satisfied by fewer groups) as $$c$$ increases, while condition (2) gets less demanding (i.e., weaker, so satisfied by fewer groups) as we increase $$c$$. Thus, a particular value of c may work for a Lie ring but larger and smaller values may not.

A Lazard Lie ring is a Lie ring that can participate as the Lie ring side of a Lazard correspondence. The other side of the correspondence is its Lazard Lie group.

p-Lie ring
A Lazard $$p$$-Lie ring is a special case of the above, namely a Lie ring $$L$$ such that:


 * 1) There is a prime $$p$$ such that every element of $$L$$ has order a power of $$p$$.
 * 2) The Lie subring of $$L$$ generated by any three elements of $$L$$ is a nilpotent Lie ring of nilpotency class less than $$p$$.

A Lazard $$p$$ Lie ring is a $$p$$-Lie ring that can participate as the Lie ring side of a Lazard correspondence. The other side of the correspondence is its Lazard Lie group, which is a p-group.

Stronger properties

 * Weaker than::Abelian Lie ring

Weaker properties

 * Stronger than::Nilpotent Lie ring