Nilpotent elements and nil right ideals

Question

If $x_1,\dots x_n$ are nilpotent elements in a unital ring $R$, is it true that their sum falls into the sum of all nil right ideals of $R$?

In fact I guess somehow that any nilpotent element in a unital ring falls into a nil right ideal of $R$.

Answer 1

Obviously not, in both cases.

A matrix ring over a field ($n>1$, of course) has no nontrivial nil right ideals, but it still has nilpotent elements.