Graded Lie algebras with few nontrivial components

We prove that if a $(\mathbb Z/n\mathbb Z)$-graded Lie algebra $L=\bigoplus\limits_{i=0}^{n-1}L_i$ has $d$ nontrivial components $L_i$ and the null component $L_0$ has finite dimension $m$, then $L$ has a homogeneous solvable ideal of derived length bounded by a function of $d$ and of codimension bounded by a function of $m$ and $d$. An analogous result holds also for the $(\mathbb Z/n\mathbb Z)$-graded Lie rings $L=\bigoplus\limits_{i=0}^{n-1}L_i$ with few nontrivial components $L_i$ if the null component $L_0$ has finite order $m$. These results generalize Kreknin's theorem on the solvability of the $(\mathbb Z/n\mathbb Z)$-graded Lie rings $L=\bigoplus\limits_{i=0}^{n-1}L_i$ with trivial component $L_0$ and Shalev's theorem on the solvability of such Lie rings with few nontrivial components $L_i$. The proof is based on the method of generalized centralizers which was created by E. I. Khukhro for Lie rings and nilpotent groups with almost regular automorphisms of prime order [1], as well as on the technique developed in the work of N. Yu. Makarenko and E. I. Khukhro on the almost solvability of Lie algebras with an almost regular automorphism of finite order [2].