Given a z transform with one pole can you have a 2 sided Region of convergence or does 1 pole limit it to being only left or right sided? I know when you have two poles the 2 sided scenario is when a "ring" forms between the two poles in the z-space but when you have one pole does the case when the entire z space is in the RoC excluding the ring formed by the one pole count as the two sided case?
With one pole you have only two possible regions of convergence, either $|z|<r$ (where $r$ is the pole radius), which corresponds to a left-sided sequence, or the region $|z|>r$, 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.
What nobody ever explains is why you need the two sided z transform in the first place! Now usually in control systems all systems are causal so nothing exists in negative time anyway. Now of course for optimal control the maths needs positive and negative sequences but for most problems of ordinary engineering we don't. A causal s domain means causal z. I suppose once again in optimal wiener filters we come across two sided z as well.