Some Varieties of Lie Rings

In this paper, several theorems of Macdonald \cite{Mac1961,Mac1962} on the varieties of nilpotent groups will be generalized to the case of Lie rings. We consider three varieties of Lie rings of any characteristic associated with some equations (see Eqs. (\ref{eq:1.1})-(\ref{eq:1.3}) below). We prove that each Lie ring in variety $(\text{ref}eq:1.1)$ is nilpotent of exponent at most $n+2$; if $L$ is a Lie ring in variety $(\text{ref}eq:1.2)$, then $L^2$ is nilpotent of exponent at most $n+1$; and each Lie ring in variety $(\text{ref}eq:1.3)$ is solvable of length at most $n+1$. Finally, we also discuss some varieties of solvable Lie rings and the varieties of Lie rings defined by the properties of subrings.