Vertically coupled double quantum rings submitted to a perpendicular magnetic field $B$ are addressed within the local spin-density functional theory. We describe the structure of quantum ring molecules containing up to 40 electrons considering different inter-ring distances and intensities of the applied magnetic field. When the rings are quantum mechanically strongly coupled, only bonding states are occupied and the addition spectrum of the artificial molecules resembles that of a single quantum ring, with some small differences appearing as an effect of the magnetic field. Despite the latter has the tendency to flatten the spectra, in the strong coupling limit some clear peaks are still found even when $B\neq 0$ that can be interpretated from the single-particle energy levels analogously as at zero applied field, namely in terms of closed-shell and Hund's-rule configurations. Increasing the inter-ring distance, the occupation of the first antibonding orbitals washes out such structures and the addition spectra become flatter and irregular. In the weak coupling regime, numerous isospin oscillations are found as a function of $B$.
