  We study certain (two-sided) nil ideals and nilpotent ideals in a Lie
nilpotent ring R. Our results lead us to showing that the prime radical rad(R)
of R comprises the nilpotent elements of R, and that if L is a left ideal of R,
then L+rad(R) is a two-sided ideal of R. This in turn leads to a Lie nilpotent
version of Cohen's theorem, namely if R is a Lie nilpotent ring and every prime
(two-sided) ideal of R is finitely generated as a left ideal, then every left
ideal of R containing the prime radical of R is finitely generated (as a left
ideal). For an arbitrary ring R with identity we also consider its so-called
n-th Lie center Z_n(R), which is a Lie nilpotent ring of index n. We prove that
if C is a commutative submonoid of the multiplicative monoid of R, then the
subring of R generated by the union of Z_n(R) and C is also Lie nilpotent of
index n.
