  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 $B$.
