Broken Pull chain light

The metal ring is there to keep the chain from going too far into the fixture, and yes, that's the connecting point for your chain.

If you're worried about the electricity, just turn it off while you do the repair, but it shouldn't be necessary. I'd worry more about you falling or standing on something unsafe while you're doing it, if the light socket is open with no bulb and you don't turn the power off, then yes there is a shock hazard... which would be you somehow reaching inside the socket with a finger... Murphy's law says: Yes, turn off the power first. if you put a bulb in it could make it harder to get at it, if the light's on it could get hot and burn you... life's full of openings for Murphy so go ahead and turn the power off (with a breaker or the main power at the fuse-box). shouldn't be difficult. Sometimes they get stuck, or the switch mechanism inside wears out, both of those can be fixed if necessary. Should you run into either of those problems holler for more help. But unless there's more than a broken chain wrong, it should be a piece of cake. Stay safe...

and Good Luck.