ballast wiring

[bv_reefer]

Ok so I went and got my mogul socket and socket bracket that holds it together. The cable that's gonna connect to my socket has 3 exposed wires, a black wire, white wire, and green wire. the socket has a black one and a white one.

So just wanted to double check here, white wire with white wire, black with black, twist together and twist them together with a wire connector. correct?

Next, theres a green wire on the cable coming from my ballast, and theres also a green one attached to the aluminum socket bracket, ( grounding wires i believe?) so do I join these together or do I take off a centimeter or so of insulation and put a wire connector on them individually so there not exposed? correct?