On one-sided Lie nilpotent ideals of associative rings

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 $[\ldots[ [r_1,\,r_{2}],\ldots],\,r_{n-1}],\,r_{n}]$ are also studied.