Changes of signs in conditionally convergent series on a small set

We consider ideals I of subsets of the set of natural numbers N such that for every conditionally convergent series of real numbers ∑ n ∈ N a n and s ∈ R ¯ , then there is a sequence of signs δ = ( δ n ) n ∈ N such that ∑ n ∈ N δ n a n = s and N ( δ ) : = { n ∈ N : δ n = − 1 } ∈ I . We give some properties of such ideals and characterize them in terms of extendability to a summable ideal.