Kreknin's theorem on existence of Kreknin's function

Statement
There exists a function $$k$$, is a function from the set of natural numbers to the set of natural numbers that satisfies the following:

Any $$\mathbb{Z}/n\mathbb{Z}$$-graded Lie ring (i.e., a Lie ring graded over the finite cyclic group of order $$n$$; see Lie ring graded over an abelian group) with degree zero component equal to the zero Lie ring has derived length at most $$k(n)$$.

The smallest possible such function is termed Kreknin's function.

Facts used

 * 1) uses::Subvariety of variety of Lie rings containing no nontrivial perfect Lie rings contains only solvable Lie rings (part of this statement is also that there is a common bound on the derived length of all the Lie rings in the subvariety)

Abstract proof
This version of the proof does not provide explicit upper bounds on Kreknin's function.

Proof with concrete upper bound on Kreknin's function
This version of the proof shows that $$k(n) \le 2^{n - 1}$$.