When I turn the A/C On, RPM fall down and the battery lights turns on.

have the battery checked and the alternator.one of them is bad or both.you must have a full charge on the battery before alternator can be checked while on vehicle,unless you remove it and take to autozone,or advance.if you do this way, stand near testing machine display and watch to make sure of results.have them test it at least 3 times. it sounds like the alternator is bad if car is cutting off,but if battery is bad,it will not stay maintained by alternator.the battery is for starting the car,then alternator takes care of running all electrical including computer,pcm,ecm ,etc.check the sides of the battery,there should be a round sticker on it with the year it was bought.average battery life now is 4 years,date could be on top of battery.heat is what kills a battery.its probably got a group 36R BATTERY IN IT.

It could be the Battery not holding charge well.. but I'm leaning more towards the alternator because there was no mention about battery light coming on for any other reason..even with the weak battery the alternator should provide sufficient power to keep the battery light from coming on during a/c operation. have the alternator checked along with the battery