I dropped a ring into the lint trap while the trap was out; how can I retrieve the ring?

You will be needing to take the dryer apart to find the "something".
God bless your efforts.

if there is no water any where I would first use the wand of a vacuum cleaner that is going ( not the head just the pipe section)

If it's a metal utensil that responds to magnetism, try a magnet on a string, or one of those telescoping magnetic picker-upper tools to get it out.

You can remove the lint filter to retrieve the RING, please do not put the machine to any wash until the filter and drain pump is removed and checked.
check this link.; Cleaning lint filter

take the lint screen out and turn on the dryer,, holding the lint screen door open, it should just blow out, if not,, you have to unplug the unit, remove the back,, and get into the lint trap , if you can see the paper towel,, auto parts stores have a long pick up tool that has Little prongs at the end,, and you can reach down with it and grab the thing !!

its not too hard to get out.
you will need a #2 phillips screwdriver and or a 1/4" socket driver
remove lint trap
remove 2 screws at lint trap
unplug unilt
open the top (2 clips between lid and front panel in the crack, push them with a butter knife or putty knife to release top) opens like a hood
pull out dryer and pull off rear panel
on the left hand side there is a metal duct like thing lol
that leads to the lint trap at the top.
there should be 4 screws at the bottom where it gets round.
remove them, and wiggle the duct open. your pen should be there..

You can gain access or remove the lint trap by popping the top of the dryer up, then remove the front panel.

Surely you should be able to remove lint trap, or you would not be able to clean it!!!!.
Otherwise try a piece of wire with a hook on the end , and retrieve it that way.

is the lint trap on top of or inside the door?