The alternator light is the "indicator" in a voltage balance circuit --
it says there is current flowing through the alternator winding, not
"balanced" by output. The circuit is usually not directional, so
current flow from the battery to the alternator will cause it to come
on (not chargine), but so will current flow from a short. On older
cars with non-electronic ignitions, it could cause "runon"-- enough
current would feed back through the low output from the alternator to
keep the ignition firing with the key off!
The result is that so long as you have at least 12V output on the
alternator, the lamp is out, even if the battery is actually
discharging. You will also get the lamp on if the battery is
overcharging sometimes, too due to a failed winding (not a regulator),
with current going through the ignition switch to the battery rather
than the other way round.
If it's on with the key off, you have an internal short in the
alternator or regulator, hence your dead battery.
Check the battery too -- alternator and battery failures can be
interrelated, with a bad alternator roasting a battery, or a bad
battery overloading and frying the alternator, and if the battery is
bad, you may be killing a new alternator pretty fast!
Peter