Interesting fileā¦ Anyways, I found that turning on āztranspā (under links and pipelines in the materials settings) helps, but for some reason doesnāt seem to fix the problem. Neither does āraytranspā.
Havenāt gone into it, but itās something to do with your render layer to node setup. When I disabled ādo compositeā and rendered your first renderlayer (and activated the layer on which your rings sit) it worked fine.
Maybe you can revisit your render layers and how they flow into your node setup, specifically paying attention to the AllZ button and how you are combining the different layers with nodes.
Hope this helps!
Iām at a library so I canāt play with your file but if youāre trying to use a z-combine node you need to be aware that this is not compatible with alpha channels. It is basically a front and back node so anything combined behind a z transparent face will be occluded. The AllZ button that freen mentioned is a much better option because it reduces file sizes, render times and does not require recombination via a z-combine nodeā¦simply use an Alpha Over node.