Stable elements and property (S)

We study the relation (and differences) between stability and Property (S) in simple stably finite framework. This leads us to characterize stable elements terms of their support, these concepts from different sides: hereditary subalgebras, projections multiplier algebra order properties Cuntz semigroup. use approaches show both that cancellation at infinity on semigroup just holds when its equivalence is given by isomorphism level Hilbert right-modules, notions such as Regularity, ω-comparison, Corona Factorization Property, property R, etc. are equivalent under mild assumptions.