I did something that hurt my ex pretty bad, and we went through a rough breakup. Would it be a good idea to send my ex an apology letter, and do you think we could still be friends?
Received via email (name withheld)
My personal take on apologizing after a breakup is that it usually helps both parties with closure. The key thing is including all the components of a good apology (Taking ownership of what you did wrong, expressing that you understand the negative effect your actions had on the person whose trust you broke, and expressing a desire to make amends / correct the situation going forward). In the case of a breakup, correcting the situation going forward really means emphasizing that you hope the person won't get harmed similarly in the future, and that you will no longer be harming the person.
Definitely emphasize that you aren't sending the letter / email in hope of getting back together, but just to create better closure on the situation. I do recommend that you send a letter / email rather than call or instant message, as you don't want to create an opportunity for your ex to argue, express recriminations, or lash out at you.
As far as being friends again goes, that may be possible down the line, but you probably don't want to express a desire to be friends again in the apology letter, for two reasons. Firstly, if your ex is still hurting, he or she likely won't be too receptive; give it more time. Secondly, if you ask to be friends as part of the apology, it may come across as self-serving (i.e. the letter isn't so much to admit fault and make your ex feel better, but is instead to manipulate your ex into being friendly again). See how your ex takes the letter, give it more time, and then after a bit more time has passed, you might consider reaching out again to see if your ex is receptive to being friends.