  We prove that a Lie nilpotent one-sided ideal of an associative ring $R$ is
contained in a Lie solvable two-sided ideal of $R$. An estimation of derived
length of such Lie solvable ideal is obtained depending on the class of Lie
nilpotency of the Lie nilpotent one-sided ideal of $R.$ One-sided Lie nilpotent
ideals contained in ideals generated by commutators of the form $[... [ [r_1,
r_{2}], ... ], r_{n-1}], r_{n}]$ are also studied.
