Hi, I'd like to help you with your question.
It sounds as if you and your mother were both hurt by this argument. She is using her hurt as a way to control the situation and make you apologize so she will feel better. Although it would be ideal for you both to make amends together, she may not be willing to do so.
You have a few choices here. You can ignore her and move on. From what you said, this option doesn't sound like a workable one for you. It is also not the best idea if you care for your mother. However, it is an option.
Second, you can let her know you are willing to apologize but she needs to as well. Let her know you were hurt by the argument too, and that you feel both of you were at fault. She may not agree and continue with how she is acting, or she may give in. In that case, you both should meet on neutral ground, say for lunch, and talk it out. This way, it will be more difficult to argue again since you will be in a public place and also a neutral place, where neither of you have the advantage.
Third option is to apologize to her. Yes, it feels like acknowledging she is right and it is humbling to do so, but it may be necessary if you would like to continue to maintain your relationship with her. Sometimes you have to be the better person and give a small token for the benefit of the bigger picture. This way, she gets what she wants and you get your relationship back.
I hope this has helped you,