Inner and outer rings are not strongly coupled with stellar bars

We study the frequency and dimensions of inner and outer rings in the local Universe as a function of disk parameters and the amplitude of non-axisymmetries. We use the 1320 not-highly inclined disk galaxies ($i<65^{\circ}$) from the S$^4$G survey. The ring fraction increases with bar Fourier density amplitude: this can be interpreted as evidence for the role of bars in ring formation. The sizes of inner rings are positively correlated with bar strength: this can be linked to the radial displacement of the 1/4 ultra-harmonic resonance while the bar grows and the pattern speed decreases. The ring intrinsic ellipticity is weakly controlled by the non-axisymmetric perturbation strength: this relation is not as strong as expected from simulations, especially when we include the dark matter halo in the force calculation. The ratio of outer-to-inner ring semi-major axes is uncorrelated with bar strength: this questions the manifold origin of rings. In addition, we confirm that i) $\sim 1/3$ ($\sim 1/4$) of the galaxies hosting inner (outer) rings are not barred; ii) on average, the sizes and shapes of rings are roughly the same for barred and non-barred galaxies; and iii) the fraction of inner (outer) rings is a factor of $1.2-1.4$ ($1.65-1.9$) larger in barred galaxies than in their non-barred counterparts. Finally, we apply unsupervised machine learning (Self-Organizing Maps, SOMs) to show that, among early-type galaxies, ringed or barred galaxies cannot be univocally distinguished based on 20 internal and external fundamental parameters. We confirm, with the aid of SOMs, that rings are mainly hosted by red, massive, gas-deficient, dark-matter poor, and centrally concentrated galaxies. We conclude that the present-day coupling between rings and bars is not as robust as predicted by numerical models (Abridged).