I think it is best for you to not talk to her anymore and both move on with your lives. Keeping in contact will only hurt and cause feelings of anger and jealousy as one person moves on quicker than the other. Also, if a full fledged relationship didn't work out and you were fighting all the time, then what's the point of "remaining" friends? Hell, friends don't fight all the time....
We have all been in these situations and its very hard; but each person handles it in his own fashion. I would suggest you cut all contact with her and fill your life up with new projects, goals, ideas, plans, activities, anything that makes you feel good. Take up new hobbies, new sports, take classes, etc etc, move on with your life and set your life up in such a fashion that you wouldn't even have time to be her friend even if she wanted to.
Also, you should start focusing on new women and new relationships. Learn the stuff on this site, go out there and approach women, practice, have fun, and before you know it, you'll realize that that breakup was the best thing that ever happened to you.
Dude, I want to read your field reports, I want to read your lay reports, I want you to kick a$$ at this stuff. Forget her and plunge right in to new experiences and new women.
Best of luck.
Cesare Cardinali