2-sided Regions of convergence for Z transforms
With one pole you have only two possible regions of convergence, either $|z|&lt;r$  (where $r$ is the pole radius), which corresponds to a left-sided sequence, or the region $|z|&gt;r$, corresponding to a right-sided sequence. I think the misunderstanding lies in the concept of 'the ring formed by the one pole', because there is no such ring.