Don, your mother was right -- funerals ARE for the living. For many, it is a necessary part of the grief process. I agree with MaryAnn That a small memorial service at an appropriate place would suffice. If you don't have something, the people still living are denied the chance to celebrate and remember your mom and comfort each other.
There was a woman at church who did not have a service for her husband. Everyone else felt awful. So, the pastor suggested a small memroial service for everyone else and that's what the friends did. So, in that case, I think everyone got their needs met.
As for her ashes, if she did not specify what she wanted done with them, then it is up to you as next of kin to make those decisions. Hang in there. Don