  We study the fragmentation of collisional ring galaxies (CRGs) using a linear
perturbation analysis that computes the physical conditions of gravitational
instability, as determined by the balance of self-gravity of the ring against
pressure and Coriolis forces. We adopt our formalism to simulations of CRGs and
show that the analysis can accurately characterise the stability and onset of
fragmentation, although the linear theory appears to under-predict the number
of fragments of an unstable CRG by a factor of 2. In addition, since the
orthodox `density-wave' model is inapplicable to such self-gravitating rings,
we devise a simple approach that describes the rings propagating as material
waves. We find that the toy model can predict whether the simulated CRGs
fragment or not using information from their pre-collision states. We also
apply our instability analysis to a CRG discovered at a high redshift,
$z=2.19$. We find that a quite high velocity dispersion is required for the
stability of the ring, and therefore the CRG should be unstable to ring
fragmentation. CRGs are rarely observed at high redshifts, and this may be
because CRGs are usually too faint. Since the fragmentation can induce active
star formation and make the ring bright enough to observe, the instability
could explain this rarity. An unstable CRG fragments into massive clumps
retaining the initial disc rotation, and thus it would evolve into a clumpy
galaxy with a low surface density in an inter-clump region.
