We study the fragmentation of collisional ring galaxies (CRGs) using ourlinear perturbation analysis that computes the physical conditions ofgravitational instability. We adopt the analysis to our CRG simulations, andshow that our analysis accurately characterises the stability and the onset offragmentation that is determined by the balance of self-gravity againstpressure and Coriolis forces. In addition, since the orthodox `density-wave'model is inapplicable to such self-gravitating rings, we devise a simple modelthat describes the rings propagating as material waves. We find that the toymodel can predict the behaviour of the rings in our simulations. We also applyour instability analysis to a CRG discovered at a high redshift $z=2.19$. Wefind that a quite high velocity dispersion is required for the stability of thering, and therefore the CRG can be unstable against ring fragmentation. CRGsare rarely observed at high redshifts, and it may be because CRGs are usuallytoo faint. Since the fragmentation can induce active star formation and makethe ring bright enough to observe, the instability may explain the rarity.Fragmenting CRGs would evolve into clumpy galaxies with low surface densitiesin their inter-clump regions.