Rearrangement of conditionally convergent series on a small set

We consider ideals I of subsets of the set of natural numbers such that for every conditionally convergent series ∑ n ∈ ω a n and every r ∈ R ¯ there is a permutation π r : ω → ω such that ∑ n ∈ ω a π r ( n ) = r and { n ∈ ω : π r ( n ) ≠ n } ∈ I . We characterize such ideals in terms of extendability to a summable ideal (this answers a question of Wilczyński). Additionally, we consider Sierpiński-like theorems, where one can rearrange only indices with positive a n .