Saeed bin Firoz narrated from his father that Sayyiduna Umar (may Allah be pleased with him) said regarding the issue of a polytheist woman: The father and closeness create a bond among all three types of brothers.
Brief Explanation
(Explanation of Hadiths 2916 to 2921)
From all these traditions, it is understood that Sayyiduna Umar, Sayyiduna Uthman, Sayyiduna Zayd (radi Allahu anhum), Qadi Shurayh (rahimahullah), and others would include all brothers—whether full, consanguine (sharing the same father), or uterine (sharing the same mother)—as heirs, whereas Sayyiduna Ali (radi Allahu anhu) would not include them.
The scenario of the issue is as follows: If a woman passes away leaving behind a husband, mother (or grandmother), uterine brothers, and one or more full brothers, then in this case, the inheritance will be divided into 6 shares. Of these, half, i.e., 3 shares, will go to the husband; one-sixth, i.e., one share, to the mother; and one-third, i.e., two shares, to the uterine brothers. The full brothers, because they are ‘asaba (agnatic heirs), will receive nothing, as when the shares of the fixed-share heirs (ashab al-furud) are completed, the ‘asaba are deprived.
Husband . . . . half . . . . 3
Mother . . . . one-sixth . . . . 1
Uterine brother . . . . one-third . . . . 2
Full brother . . . . nothing
However, Amir al-Mu’minin Sayyiduna Umar (radi Allahu anhu), through his ijtihad (independent juristic reasoning), decreed that the full brother or brothers should be included in the one-third share of the uterine brothers. Therefore, all will divide this share equally, and this is a specific case in which the full brother is treated like the uterine brother, and in the absence of other heirs, all will share in the inheritance. This issue is also called mushtaraka, mushtaraka hijriyya (and also ‘umariyya and himariyya).