A self-stabilizing system has the property that, no matter how it is perturbed, it eventually returns to a legitimate configuration. Dijkstra originally introduced the self-stabilization problem and gave several solutions for a ring of processors in his 1974 Communications of the ACM paper. His solutions use a distinguished processor in the ring, which effectively acts as a controlling element to drive the system toward stability. Dijkstra has observed that a distinguished processor is essential if the number of processors in the ring is composite. We show, by presenting a protocol and proving its correctness, that there is a self-stabilizing system with no distinguished processor if the size of the ring is prime. The basic protocol uses Θ ( n 2 ) states in each processor when n is the size of the ring. We modify the basic protocol to obtain one that uses Θ ( n 2 /ln n ) states.