Lazard Lie ring of adjoint group of a radical ring equals associated Lie ring of the radical ring under suitable nilpotency assumptions

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Statement

Adjoint group version

Suppose N is a radical ring such that there exists a positive integer n so that the additive group of N has powering threshold at least n - 1 and x^n = 0 for all x \in N. Suppose N has adjoint group G. Suppose that G is a Lazard Lie group. Then, the Lazard Lie ring of G is isomorphic to the Lie ring associated to N, i.e., the Lie ring whose additive group is the same as that of N and whose Lie bracket is the additive commutator in N, i.e., [x,y] := xy - yx.

Algebra group version

Suppose q is a power of a prime p and N is a nilpotent associative algebra over the finite field \mathbb{F}_q. Suppose, further, that x^p = 0 for all x \in N.

Let G be the algebra group corresponding to N. Then, G is a Lazard Lie group. Then, the Lazard Lie ring of G is isomorphic to the Lie ring associated to N, i.e., the Lie ring whose additive group is the same as that of N and whose Lie bracket is the additive commutator in N, i.e., [x,y] := xy - yx.