We present and analyse Las Vegas distributed algorithms which compute a MIS or a maximal matching for anonymous rings. Their bit complexity and time complexity are $O(\sqrt{\log n})$ with high probability. These algorithms are optimal modulo a multiplicative constant. Beyond the complexity results, the interest of this work stands in the description and the analysis of these algorithms which may be easily generalised. Furthermore, these results show a separation between the complexity of the MIS problem (and of the maximal matching problem) on the one hand and the colouring problem on the other. Colouring can be computed only in $\Omega(\log n)$ rounds on rings with high probability, while MIS is shown to have a faster algorithm. This is in contrast to other models, in which MIS is at least as hard as colouring.