I have always said sorry to my mother.

Talk, don't fight. It accomplishes nothing. Go ahead and tell her you are sorry for arguing with her. If you think you are right on the issue, talk with her some more about it.

The person at-fault should make the first move to apologize.

You only have one mother. if you were wrong, tell her so.

She will not be on earth forever.

Always say sorry first.
I had trouble before not apologizing.
I was too proud and my pride just made things worse.
But now, I see that I need to humble myself every now and then. :D
It will teach others to do the same.
I hate that feeling when you're pissed off at each other. It's unnecessary and sickening.
But whenever I have some problems with my mom I tell her I want to explain myself. Sit down and TALK it through. Tell her what you think you did wrong and what she did wrong, CALMLY. Say, what you want to change next time the same thing happens, how you should handle it.
Hope all goes well :D
God bless