the battery is on but the car keep starting could that be the alternator?

if you mean the battery LIGHT is on,then take the car and have the charging system tested at Autozone.it's free!if the alternator is bad,then you are running your car off of the battery,until it drains it,then it will die on you and strand you somewhere!battery MUST have a full charge on it to check the alternator while still mounted on car.