Summary: We analytically work out the long-term orbital perturbations induced by a homogeneous circular ring of radius \(R_r\) and mass \(m_r\) on the motion of a test particle in the cases (I): \(r>R_r\) and (II): \(r<R_r\). In order to extend the validity of our analysis to the orbital configurations of, e.g., some proposed spacecraft-based mission for fundamental physics like LISA and ASTROD, of possible annuli around the supermassive black hole in Sgr A* coming from tidal disruptions of incoming gas clouds, and to the effect of artificial space debris belts around the Earth, we do not restrict ourselves to the case in which the ring and the orbit of the perturbed particle lie just in the same plane. From the corrections \(\Delta\dot\varpi^{(\mathrm {meas})}\) to the standard secular perihelion precessions, recently determined by a team of astronomers for some planets of the Solar System, we infer upper bounds on \(m_r\) for various putative and known annular matter distributions of natural origin (close circumsolar ring with \(R_r = 0.02 - 0.13\) au, dust ring with \(R_r= 1\) au, minor asteroids, Trans-Neptunian Objects). We find \(m_{\mathrm r}\leq 1.4\times 10^{-4}\, m_{\oplus}\) (circumsolar ring with \(R_r = 0.02\) au), \(m_{\mathrm r}\leq 2.6\times 10^{-6}\, m_{\oplus}\) (circumsolar ring with \(R_r= 0.13\) au), \(m_{\mathrm r}\leq 8.8\times 10^{-7}\, m_{\oplus}\) (ring with \(R_r = 1\) au), \(m_{\mathrm r}\leq 7.3\times 10^{-12}\, M_{\odot}\) (asteroidal ring with \(R_r = 2.80\) au), \(m_{\mathrm r}\leq 1.1\times 10^{-11}\, M_{\odot}\) (asteroidal ring with \(R_r = 3.14\) au), \(m_{\mathrm r}\leq 2.0\times 10^{-8}\, M_{\odot}\) (TNOs ring with \(R_r = 43\) au). In principle, our analysis is valid both for baryonic and non-baryonic Dark Matter distributions.