I take a really coarse wash cloth, wet it thoroughly, put in some bar soap scraps and wrap it like a scouring pad, then use it to scrub the ring around the tub. Then I just throw it in the laundry when I'm done. If it's a hot (whites) wash, and you don't use too many slivers (guess how I found THAT out...), then it will dissolve in the wash just fine. :)
You can call me Anika. :)