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