Slicing a disc into two or more is a good way to make up thin rings. The only way to do this safely is to use a jig of some sort. Gord Rock has posted a video showing a very simple way.
View attachment 2115 Here's a picture of adapting a mortising jig to re-saw work. Picture shows re-sawing for a dizzy bowl but works as well for splitting rings. Full disclosure, not my idea. Got it from Wayne Miller.