No rings in the TV series (just the games), and no Chaos Emeralds or Super Sonic either!
My one major beef with Boom is that despite being a brand new universe, they have done very little world building. No explanations for how Sonic and friends came to meet each other, no backstory for Knuckles, no reason why Eggman is a human living off the coast of an animal island. This sort of stuff is what I loved about the classic games, and I was really hoping to see a new take on that sort of stuff.
Imagine if a new TMNT series released and they didn't detail how this new version's team came to be or who Splinter was or what Shredder's history with the turtles was.