I have two questions:
1) If $S$ is the sum of all right nil ideals of a ring $R$ (with unity), is it true that $S$ is a two-sided ideal? It is clear for me that $S$ is a right ideal (and it is nil if and only if Koethe Conjecture is valid for $R$).
2) Does the sum of all nil two-sided ideals of $R$ contain every nil right ideal?
Thanks!
1) $S$ is also a left ideal, in fact, if $r\in R $ and $I$ is a right ideal of $R$ so is $rI$ (Evident!). Let $x\in I$, we first observe that $(xr)^k=0$ for some integer $k$. Therefore, we have $(rx)^{k+1}=0$, showing that $rI$ is a nil right ideal of $R$. Now, if $y\in S$ we can write $y=y_1+...y_n$, where each $y_j$ belongs to some nil right ideal $I_j$ of $R,$ for$ j=1,...,n$. For any $r\in R$ the element $ry$ would be a sum of $n$ elements each of which falls into some nil right ideal of $R$. So, $ry\in S$.
2) This is one of the statements equivalent with Koethe Conjecture.