  We introduce the notion of solid monoid and rigid monoid in monoidal
categories and study the formal properties of these objects in this framework.
We show that there is a one to one correspondence between solid monoids,
smashing localizations and mapping colocalizations, and prove that rigid
monoids appear as localizations of the unit of the monoidal structure. As an
application, we study solid and rigid ring spectra in the stable homotopy
category and characterize connective solid ring spectra as Moore spectra of
subrings of the rationals.
