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The nut broke off and had a small red ring/gasket between these connectors. I'm an noob but can follow instructions. Please give suggestions. Thanks
Below there is a water valve 
which may be used as an alternative if the above (cupper?) pipe is inconvenient to use.
I think one of these would work for you.
https://g.co/kgs/yJgKbST
Do those bite to copper?
Yes. Sharkbite connectors are great for fitting new and old plumbing together.