For this example I made a little doodle that starts with the middle ring of a clover.
R: 3+ 3-5.
Joined the chain to the base of the rings.
Now cut the ball thread, leaving around 3 inches or what ever amount it takes to reach to the middle of the next ring. I hate to waste thread so for me 2-3 inches is fine.
I encapsulate for 5 stitches; just reaching to the ds before making the picot. I don't like to cut ends on a picot because it disfigures the picot.
After making the picot, I start my SSSR. To make a SSSR go to this tutorial.
Then you can cut both ends close to the ds. If you're afraid that the ring may open, you can put a drop of fabric glue on the base of the last ring, where the loop of the SSSR is anchored.
That's all! In this way you can finish your ends without having to use a needle to sew ends or use a filament for the magic thread or what ever.
This is showing how to do it with a work ending on a ring that has just one joined side. Can be used also when the ring stands alone.
For a ring that is joined on both side you will need to adjust the SSSR. Ladytats shows how to do this.