Are you asking about something based on smartthings, because the biggest difference between smartThings and any UL listed security system, including ring, is that smartthings is not UL listed for security because among other things it’s very cloud dependent. It cannot send any notifications, not even to your own phone, unless the Internet is working and the smartthings cloud is available. There isn’t even a way to arm and disarm it without the Internet.
So many people in this community do have a separate security system and just use smartThings for home automation because, to be honest, the notification capability just isn’t sufficient for a primary security system.
That’s not just my opinion: the company says so themselves in their product usage guidelines:
https://www.smartthings.com/guidelines
Data accuracy and consistency from SmartThings sensors, including those provided by SmartThings directly, resold by SmartThings, or supported by SmartThings, is not guaranteed. Therefore, you should not rely on that data for any use that impacts health, safety, security, property or financial interests.
So while there is some redundancy, if you truly need a security system, the ring is a good choice and priced low enough that the redundancy is usually affordable.
None of the UL listed security systems will integrate with smartthings in a way that allows you to use the smartthings home automation sensors to trigger an alarm. Because of the issues raised in the product usage guidelines.
Even The dual logo ADT/SmartThings system (which appears to be in end-of-life stage, and so it’s very hard to find and cannot really be recommended anymore) didn’t use the regular smartThings sensors: it introduced its own sensors on a different frequency for the security options.
So it sounds like you’ve made all the right choices so far if you like smartthings for your home automation system. Getting a separate security system is really best practices at this point, and the ring system is a good budget choice.
BTW, you can use Alexa routines (not SmartThings routines) to get partial integration between the ring security system and smartthings. You could have a ring sensor trigger an event in SmartThings if you like. But of course that integration will be cloud dependent, so just depends on exactly what you want to do as to whether that meets your requirements.