All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
It is not permissible to prevent any one of those sons from his share in the inheritance of his mother because of him having cut ties with her. Since it is not reported in the Sharee’ah that cutting ties with kinship is an impediment that prevents a person from inheritance, then it is not permissible for you to consider it as such; for more benefit, please refer to Fatwa 153619.
If this mother does not have any heirs other than her six sons, then they should divide her wealth between them equally and none of them should be favored (given a greater share) due to his dutifulness and kindness to his mother nor should he be given a smaller share than what he deserves because of his undutifulness toward his mother.
If there is a will or debts (left by your mother), then the will should be fulfilled and the debts should be paid before the division of the inheritance. Allaah Says (what means): {…after any bequest they [may have] made or debt.} [Quran 4:11]
It should be noted that deserting the mother is considered as a kind of cutting ties with her, and cutting ties with the parents is a great major sin especially with the mother as the Sharee’ah enjoined us to allot three-quarters of our kindness and dutifulness to her. When the Prophet was asked "Who is most deserving of my good company? he replied: "Your mother" The man said: "Then, who? The Prophet replied: "Your mother". The man again asked: "Then, who?” He [the Prophet] again replied: "Your mother". The man said: "And then, who? The Prophet then replied: "Then your father". [Al-Bukhari and Muslim]
For more benefit on the dutifulness and kindness to the parents and the prohibition of cutting ties with them, please refer to Fataawa 87856, 82254, 87019.
Hence, if what you mentioned about those sons is true that they deserted their mother, then they are in great danger; so they should take the initiative to sincerely repent and perform good deeds as much as possible. Also, they should be keen on being kind and dutiful to their mother after her death by all possible kinds of good deeds; for more benefit in this regard, please refer to Fatwa 83233.
Allaah Knows best.