Lazard Lie ring
A Lazard Lie ring is a Lie ring such that there exists a natural number with both the following two properties:
|No.||Shorthand for property||Explanation|
|1||The additive group is a powered group for the set of all primes less than or equal to .||For any prime number , and any element , there is a unique element such that .|
|2||The 3-local nilpotency class is at most .||For any subset of of size at most three, the subring of generated by that subset is a nilpotent Lie ring of nilpotency class at most .|
Condition (1) gets more demanding (i.e., stronger, so satisfied by fewer groups) as increases, while condition (2) gets less demanding (i.e., weaker, so satisfied by fewer groups) as we increase . 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 special case of the above, namely a Lie ring such that:
- There is a prime such that every element of has order a power of .
- The Lie subring of generated by any three elements of is a nilpotent Lie ring of nilpotency class less than .