  Levitation of superconductors is becoming an important building block in
quantum technologies, particularly in the rising field of magnetomechanics. In
most of the theoretical proposals and experiments, solid geometries such as
spheres are considered for the levitator. Here we demonstrate that replacing
them by superconducting rings brings two important advantages: Firstly, the
forces acting on the ring remain comparable to those expected for solid
objects, while the mass of the superconductor is greatly reduced. In turn, this
reduction increases the achievable trap frequency. Secondly, the flux trapped
in the ring by in-field cooling yields an additional degree of control for the
system. We construct a general theoretical framework with which we obtain
analytical formulations for a superconducting ring levitating in an
anti-Helmholtz quadrupole field and a dipole field, for both zero-field and
in-field cooling. The positions and the trapping frequencies of the levitated
rings are analytically found as a function of the parameters of the system and
the field applied during the cooling process. Unlike what is commonly observed
in bulk superconductors, lateral and rotational stability are not granted for
this idealized geometry. We therefore discuss the requirements for simple
superconducting structures to achieve stability in all degrees of freedom.
