A nilpotent ideal in the Lie rings with automorphism of prime order

We improve the conclusion in Khukhro's theorem stating that a Lie ring (algebra) $L$ admitting an automorphism of prime order $p$ with finitely many $m$ fixed points (with finite-dimensional fixed-point subalgebra of dimension $m$) has a subring (subalgebra) $H$ of nilpotency class bounded by a function of $p$ such that the index of the additive subgroup $|L:H|$ (the codimension of $H$) is bounded by a function of $m$ and $p$. We prove that there exists an ideal, rather than merely a subring (subalgebra), of nilpotency class bounded in terms of $p$ and of index (codimension) bounded in terms of $m$ and $p$. The proof is based on the method of generalized, or graded, centralizers which was originally suggested in [E. I. Khukhro, Math. USSR Sbornik 71 (1992) 51–63]. An important precursor is a joint theorem of the author and E. I. Khukhro on almost solubility of Lie rings (algebras) with almost regular automorphisms of finite order.