We study the magnetic behavior of a finite superconducting ring in the presence of a uniform applied field directed along its axis by means of the critical-state model and the minimization of magnetic energy. We systematically study the dependence of the magnetization and the ac susceptibility upon the geometry of the ring and develop an approximate analytical expression for the case of narrow rings of any aspect ratio. Besides, we show how the critical-current density of the superconductor can be obtained from magnetization measurements and conclude that ring geometry is a very convenient one for such a purpose. In particular, we present an expression for the full penetration field in the case of finite rings, which allows us to find the value of the critical current from the value of the magnetic field at just one point on the axis of the ring.