Considering the case of homonyms processes (some processes may share the same identifier) on a ring, we give here a necessary and sufficient condition on the number of identifiers to enable leader election. We prove that if l is the number of identifiers then message-terminating election is possible if and only if l is greater than the greatest proper divisor of the ring size even if the processes do not know the ring size. If the ring size is known, we propose a process-terminating algorithm exchanging O(n log(n)) messages that is optimal.