Accretion disks around compact objects as well as the gaseous components in galaxies often have the form of a torus. To study the structure and behaviour of magnetic fields generated in such rings, a dynamo is investigated, which is working inside a torus embedded into vacuum. The equations for the kinematic αω-dynamo are written down in toroidal coordinates (see Figure 1). Besides loss of magnetic flux by Ohmic diffusion (characterized by the magnetic diffusivity D) they describe its production by the inductive effects of differential rotation and of turbulent matter, which we have chosen as ω(r) = ω′0r and α = α0 sin θ, respectively. These equations are solved by series expansion into the exponential decay modes of slender tori, which are available in analytical form. A linear homogeneous system of equations follows for the expansion coefficients; its eigenvalues determine the time-dependence of the solutions, the dynamo modes.