It is shown that a locally nilpotent ring with maximality condition for two-sided ideals is nilpotent. The restriction on the characteristic in one of the author's previously published theorems is lifted. A one-sided nil-ideal of an alternative ring, satisfying the maximality condition for right ideals, is a nilpotent ring. An example is constructed of a commutative locally nilpotent ring $A$ with maximality condition for ideals which is idempotent: $A=A^2$.