Based on the information that I have come across, the wives of the Prophet (s) did not receive an inheritance from his estate, but rather were given a 'provision' or stipend. This was either left for them in the lifetime of the Prophet (s) or was given by Abu Bakr as the first defacto caliph.
So were the wives not entitled to any inheritance? There is an interesting narration in the Mu'jam al-Buldan of Yaqut al-Hamawi that after the demise of the Prophet (s) the wives considered sending 'Uthman (presumably bin 'Affan) to Abu Bakr as their representative to demand their eighth share. But 'Aisha, the daughter of Abu Bakr and one of the wives of the Prophet (s), opposed it and all the wives withdrew from such a demand.
This is important because had the wives been entertained in their claim for inheritance, then Lady Fatima's (a) claim to Fadak could not have been refuted on the grounds that the Prophets do not leave heirs as was claimed by the first caliph Abu Bakr.
Fadak, in any case, was a lifetime gift that the Prophet (s) had made to Fatima (a). That estate was not handed over to her by Abu Bakr despite her claims to it. This is widely documented in the hadith and historical sources.
For more details on the estate of Fadak and the politics that was at play behind it being withheld from Fatima (a) please refer to:
Assalamu aleykum,
Thanks for your question.
If the son is illegitimate he has no right to inherit, so yes, the will is completely valid.
Please refer at the texts here below.
*****
- The Illegitimate Child (Walad al-Zina)
The lmamiyyah say: There is no mutual inheritance between an illegitimate child and its fornicator mother, in the same manner as there is no such inheritance between the child and its fornicator father, because there is a common impeding cause between the two, i.e. fornication. https://www.al-islam.org/inheritance-according-five-schools-islamic-law-...
***** - Views of Imamye scholars concerning inheritance of the illegitimate child
1. Most Imamye scholars assume lack of inheritance between the illegitimate child and his natural parents. In this regard Sheikh Toosi states:" child of adultery does not have lineage and does not inherit from the adulterer and the one who bore him and his heritage reaches his children and in case children do not exist, the property [inheritance] belongs to the Imam. Contemporary scholars assume lack of inheritance between the child of adultery and his parents (Khoee, 1996:414). Thus, what causes inheritance is relation and lineage where relation is connection of somebody with another by means of the latter whether relation is led to him or not (Imami 1993) http://www.worldscientificnews.com/wp-content/uploads/2015/10/WSN-44-201...
- Custody of Illegitimate Child
Jurists see that hidana and nafaqa (expenditure) of an illegitimate child must be undertaken by the child biological parents, although the child does not inherit from them. Makārim Shīrāzī, Istiftāʾāt-i Jadīd, vol. 3, p. 266.
- Imam Khomeini, Tahrirol Wasaail in English, Volume 4, Section 44: Laws on Inheritance, Fourth Impediment, birth by Fornication (Zina), page 29 onwards.
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