Should car still run on alternator with ground terminal off battery?

Just go to any AZ they will test the battery and alt. for free, in the car. It's usually safer...lol

One of the first things I said was I took it to AZ and it tested good according to them. However i feel nothing from the mag field test, and AZ Isnt exactly known for their genius "tech". I believe it was them that sold me a dead battery years ago (that I had asked them to check for charge in store before I got it home). Got it home, car still didn't start, ended up then changing a starter motor.... for nothing.