When your Laptop batteries are fully charged is it a good idea to keep the adapter on?

Remove the adapter when the battery is fully charged, you do not need to have the adapter connected all the time.

Hp etc recommend that you don't leave your power plugged in when a battery is fully charged and that if you are not using your laptop for some time you leave the battery with no more than about 10% charge in it. but this is only to prolong the life of the battery, not because of concerns of overheating the psu. as long as you sit the psu on somthing hard (not the bed or the carpet) you wont have any worries with the psu overheating - they do get quite hot in normal use. if you really want to take good care of your laptop never operate it on your lap - always use somthing solid to sit it on so it can keep itself cooled properly, always annoys me that they call them laptops - aside from dropping them overheating is the number one cause of failure.