I'm not positive but I doubt the Ring system accepts inputs from Z-Wave sensors. While most wireless alarm systems these days can control Z-Wave lights, locks and thermostats, they are only activated for an alarm by compatible wireless sensors (usually proprietary RF sensors).
Retail home automation systems on the other hand, often try and double as alarm systems and therefore will accept inputs from Z-Wave sensors.
It's a matter of Z-Wave controller class designation and Z-Wave not being reliable enough for the alarm industry to use as input sensors.