  We investigate the thermodynamic equilibrium states of a rotating thin shell,
i.e., a ring, in a (2+1)-dimensional spacetime with a negative cosmological
constant. The inner and outer regions with respect to the shell are given by
the vacuum anti-de Sitter and the rotating Ba\~{n}ados-Teitelbom-Zanelli
spacetimes, respectively. The first law of thermodynamics on the thin shell,
together with three equations of state for the pressure, the local inverse
temperature and the thermodynamic angular velocity of the shell, yields the
entropy of the shell, which is shown to depend only on its gravitational radii.
When the shell is pushed to its own gravitational radius and its temperature is
taken to be the Hawking temperature of the corresponding black hole, the
entropy of the shell coincides with the Bekenstein-Hawking entropy. In
addition, we consider simple ans\"atze for the equations of state, as well as a
power-law equation of state where the entropy and the thermodynamic stability
conditions can be examined analytically.
