Sum of Ideals of the Same Type

Question

I have two questions:

1) Is a finite sum of idempotent ideals of a ring $R$ idempotent?

2) Is any sum of nil ideals of a ring $R$ nil?

As far as I know, a finite sum of nil ideals of a commutative ring $R$ is nil too by Koethe conjecture which is true for such rings.

Answer 1

2) Every element which belongs to a sum of ideals belongs to a finite sum of them.

Answer 2

1) If $I$ and $J$ are idempotent ideals of $R$ then we have:

$(I+J)^2=I^2+IJ+JI+J^2=I+J$, since $IJ⊆I=I^2$ and also, $JI⊆J=J^2$. The general result would follow by induction.