I had a similar problem, I don't know if your car is manual or automatic, but mine is manual and the problem was the clutch.
The computer only resets when the battery is removed. The alternator just charges the battery, so if the battery is left connected when the alternator is replaced, the computer should not reset.
Could be a bad rebuild unit or a bad battery. It could also be the type of driving and accessories that the unit supports.
did you check the alternator??
Inside the alternator
the common cause why the light is on is that your alternator is not working well,try to have your alternator checked .may be the carbon brush is already worn out.
Voltage regulator or loose wire.
Check the battery cables or starter.
Alternator is not charging. Have it tested and/or replaced.
You can have the alternator tested free of charge at just about any major auto parts store. By the way, a bad alternator will not "discharge" a battery - the alternator's function is to provide the charge to the battery. If the alternator is faulty, then it is not providing the charge necessary to keep other components from discharging the battery. I had the alternator tested and it was charging. If the alternator has a bad diode, it will discharge the battery. I replaced the alternator and everything is fine.
It could be an alternator. Check with a mechanic if that is the issue.
Voltage Regulator?
alternator needs to be replaced. battery also might be a problem due to the alternator not charging it