Dish soap is the word we US'ns use for the liquid soap used to wash the dishes and dishwasher detergent is used in dishwashers.
http://homesolutionsnews.com/rbdocs/us/electrasol/
http://www.east.asu.edu/sta/u-life/housing/maint.htm

Is it just me, or does it seem that the 2nd link's writer is terribly anal about cleanliness?