  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 $(\ref{eq:1.1})$ is nilpotent of exponent at most
$n+2$; if $L$ is a Lie ring in variety $(\ref{eq:1.2})$, then $L^{2}$ is
nilpotent of exponent at most $n+1$; and each Lie ring in variety
$(\ref{eq:1.3})$ 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.
