What is the sequence type if ROC is an annulus in the z-plane if: 

$|z|&gt;|\alpha|$ is causal and right-sided

and: 

$|z|&lt;|\alpha|$ is anti-causal and left-sided

then, what is the annulus/donut-ring ROC?

$|\beta| &lt; |z| &lt; |\alpha|$ 

non-casual?  finite-length?  double-sided?
if the ROC of a Z-transform is a ring on the z-plane, then the associated time domain signal is :


two sided infinite length (hence non-causal)


and 


stable if the ring includes unit-circle or unstable otherwise.