What is the ruling on abandoning and cutting ties with one's mother after she falsely accused the wife of her honor, insulted the son, and claimed he was a miser, despite his commitment to giving her an agreed-upon monthly sum?

The right of the mother is affirmed, and honoring her is an obligation that is not waived by her ill-treatment. Therefore, you must honor her and not anger her. Try to advise her regarding her harm to you and your wife. If she responds, then praise be to Allah. If she does not respond, then the consequence of her sin is upon her. It is not permissible to abandon her or stop speaking to her, as that would make you disobedient. As for inner hatred, there is no accountability for it as long as it does not lead to disobedience. Try to overcome this hatred by remembering her past kindness.