hello

alternator not charging (faulty ) should be charging battery when car is running and producing up to 14.5 volts

look for a small blue wire on the alternator and make sure it has not broken or come off the alternator that is the wire that connects to the battery light in dash, as when engine is running current goes up the blue wire and to stop the battery bulb ignition switched current going to earth and hense the light goes out

hope that makes sense