Bob,
The Retained Accessory Power (RAP) relay permits the operation of convience accessories when the ing switch is cycled from an on to a off or lock position. Its a single pole/double throw relay. When the truck is running, the relay is denergized and the closed contacts (NC) powers devices in the truck. When you shut your truck off, this relay is energized and the other contacts close, maintaining power to various accesories until you open the drivers door. The Body Control Module (BCM) controls this relay by grounding its coil when recieving commands from the ing switch and door jamb switch.
When you shut your truck off, you should be able to hear this relay energize when you open the drivers door (turn the radio down and watch the light on the radio). If the relay is not actvating, you should be able to test the coil power and ground contacts at the relay socket with a test light.
I suspect you have a problem with the door jamb switch and/or wirring inside the door. You stated that the dome light and LH power mirror are not working. These devices are not part of the RAP circuit, but are part of the door wiring harness.
Any dealer with a scan tool can troubleshot this pretty quickly. If you have a problem with the BCM, the dealer has to change it because they have to configre the new one. On a 2001, the BCM is located below the steering column behind the "knee bolster".
MEP<br />01 2500HD, LB 4x4 Dmax w/ZF-6 spd., SBC Solid Steel Flywheel<br />DP member #739