Changes of signs in conditionally convergent series on a small set

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