When you break up with someone, it takes time for them to completely disappear from your life, if they ever do. If you move in the same social circles, share responsibilities or work together, you are going to have to talk to your ex. You don't have to have deep, meaningful conversations with former partner or spouse but you do need to be courteous and civil, whether or not you want to get back together. It is not appropriate to fight constantly with someone in public, so you should follow a few key pointers on how to talk to your ex successfully.
Talk to your ex only when necessary. "Let's stay friends" is a hollow statement. You should not ask your ex to have coffee with you for no reason; doing so won't let either of you move on. Rather, talk to your ex if you run into her or for a specific reason, such as attending to children or other mutual responsibilities. Since you have broken up, you both need to live your own lives, so your communication should be limited.
Be civil. It is extremely important for you to avoid fighting with your ex. You only get over someone when you don't have strong emotional reactions to him, so refrain from saying things that pick on him or push his buttons.
Be nice to your ex, especially if you have share responsibilities. Compliment her on things that you work on together, such as raising kids, but not on her outfit or figure because such remarks could be misconstrued as interest. Being positive will prevent fights between the two of you.
Avoid nicknames, inside jokes and other vestiges of your relationship. Your relationship is over now and you need to dispense with its trappings. This, too, will help you move forward and develop a life independent of your ex.