On solvability of Lie rings with an automorphism of finite order

A new criterion for a Lie ring with a semisimple automorphism of finite order to be solvable is proved. It generalizes the effective version of Winter's criterion obtained earlier by Khukhro and Shumyatsky and by Bergen and Grzeszczuk in replacing the ideal generated by a certain set by the subring generated by this set. The proof is inspired by the original theorem of Kreknin on solvability of Lie rings with regular automorphisms of finite order and is conducted mostly in terms of Lie rings graded by a finite cyclic group.