"Ivory dishsoap" is not "soap", its a detergent. All line manufacturers ask that detergents not be used to clean their lines. As far as I can tell -- and this is the result of more research than a sane person should admit to -- the only readily available "soap" is Ivory bar soap. Ivory "Snow" and Ivory "Flakes" used to be nothing more than the bar soap in granulated or flaked form, respectively, but now they too are detergents.
That having been said, I carry a bar of Ivory soap in a plastic bag in my fishing kit. When I want to wash my line, I put some warm water in the sink, slosh the bar of soap around in the water until I think there is enough to get the job done, put the line in the sink and wash it with a clean, soaked cloth. I then put clear water in the sink and rinse the line in the same way, dry it and, if the manufacturer suggests it for their line, treat it with "Glide" or pure silicon (Rio's product is called "Poo Goo"). SA's lines with the AST finish should be washed, but not treated with anything. Never use Armorall on any flyline.
Those are my $0.02.