How do you write a letter to my mother as to where i relinquish my rights to her inheritance?

You don't. You can let your mother know that you don't want anything, but it isn't up to you. If she has a will, she can leave you anything she wants. If she dies intestate, property would flow to you under intestate rules.

At that point, you have the right to decline any property that would otherwise go to you. You cannot do this before your mother dies.