Even if he has no contact with his ex, you may run into people when you are out together that bring up his ex or ask him questions about the divorce. Getting irritated, mad, insecure or depressed every time her name comes up is going to ruin your relationship.
Some men go through amicable divorces and still have a peaceful relationship with their ex-wives.
You don't have to be friends, but you do need to respect the fact that you are both an active part of the children's lives.
I was dating a divorced man with 3 children, he's 10 years older than I. But why wouldn't he want me included in ALL of his life, makes no sense. He started working a lot of overtime and started staying less nights with me. By this time it's been 6 hours, he can't still be there, right? Both unfortunate and fortunately, I pulled up at the same time I saw him and another woman walking out together. I couldn't be around the kids/family because he didn't really see us as a serious couple. The children were allowed to run wild at my house and that was perfectly acceptable to their father. I"m a mother too (was a single mom) and know that letting your kids rule the roost is no way to run a household.
This may be especially true if he went through an ugly divorce, he was betrayed by his wife, or if the divorce cost him a lot of money.