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

## Statement

### Adjoint group version

Suppose is a radical ring such that there exists a positive integer so that the additive group of has powering threshold at least and for all . Suppose has adjoint group . Suppose that is a Lazard Lie group. Then, the Lazard Lie ring of is isomorphic to the Lie ring associated to , i.e., the Lie ring whose additive group is the same as that of and whose Lie bracket is the additive commutator in , i.e., .

### Algebra group version

Suppose is a power of a prime and is a nilpotent associative algebra over the finite field . Suppose, further, that for all .

Let be the algebra group corresponding to . Then, is a Lazard Lie group. Then, the Lazard Lie ring of is isomorphic to the Lie ring associated to , i.e., the Lie ring whose additive group is the same as that of and whose Lie bracket is the additive commutator in , i.e., .