Should I apologize?

Ok so, long backstory.

I met my boyfriend through taekwondo. We’re both black belts. I switched to his taekwondo school two years ago, we started dating shortly after that. It’s been amazing. He’s my best friend and we have so much fun together every single day.

His ex (B) also goes to that taekwondo school. She’s also a black belt. She’s a senior like me. B and my boyfriend dated for a few months the year before him and I got together.

I’ll admit it, when I first met her I was such a bitch to her. I never said anything to her, but I did talk shit about her to my best friend (which I shouldn’t have done). I had bad relationships before this one, so I thought you were always supposed to hate your ex after the breakup. B and my boyfriend had actually broken up on good terms. They weren’t ever getting back together, but they stayed semi-friends. They never talk or text each other, but they’ll occasionally see each other at taekwondo events and they’ll hold a small conversation.

My Facebook got hacked a while back and whoever hacked it messaged B. They called her a bitch. She thought it was me and she’s hated me for it ever since.

B is one of those people that hold grudges for years. She’s hated me since my boyfriend and I got together. We think she can’t accept that he moved on and is pissed at both of us. We’d never say that to her, but it’s what we think.

Recently, I decided we should try and move away from hating each other and being childish. I’m not expecting us to be best friends or anything, I just don’t want to be hated anymore. I invited her to my graduation party. She didn’t say anything. A few weeks later, she invited my boyfriend to hers. I didn’t get an invite. Being a nice person, he messaged her and asked if he could bring me (because if he was going I wanted to go with him, not for petty reasons, I actually wanted to go) B freaked out and was like “I HATE her why would you ask me that??? She’s been such a bitch to me since you guys got together” there’s also a rumor going around that I cheated. I most definitely did not. It started a the Subway B works at because there’s a boy that looks similar to my boyfriend that works there. That boy and I used to go to taekwondo together so we know each other pretty well. We went to a dance together in like 7th grade, so he thought we dated (we didn’t) and when my boyfriend and I came in to eat one day he was telling his manager about it. I’m assuming whoever started the rumor either overheard him talking about us “dating” or they don’t know what I look like. Anyways, B also brought up the cheating rumor and said she had proof. There’s not proof because nothing happened. She “couldn’t find the proof” so she & my boyfriend stopped messaging.

Long story short:

I feel really really bad for being such a bitch to B and I want to apologize. I dont know if it’ll make things better between us, but I feel like I need to because I was such a bitch. I know she hasn’t been the nicest to me either, but I feel like she needs to hear an apology from me too.

Should I message her to apologize? I never see her in taekwondo class so I can’t do it in person.