Consider a set of mobile robots with minimal capabilities placed over distinct nodes of a discrete anonymous ring. They operate on the basis of the so called \emph{Look}-\emph{Compute}-\emph{Move} cycle. Asynchronously, each robot takes a snapshot of the ring, determining which nodes are either occupied by robots or empty. Based on the observed configuration, it decides whether to move to one of its adjacent nodes or to stay idle. In the first case, it performs the computed move, eventually. The computation also depends on the required task. In this paper, we solve both the well-known \emph{Searching} and \emph{Gathering} tasks. In the literature, most contributions are restricted to a subset of initial configurations. Here, we design two different algorithms and provide a full characterization of the initial configurations that permit the resolution of the problems under minimal assumptions.