Islamic laws of inheritance are based on a number of verses of the Qur'an and some related narrations. It is a long chapter in the Islamic law textbooks, but the following is meant to be an outline.

First, however, it is important to state that the laws of inheritance in Islam are to be taken as part of a system of laws concerning family support and financial responsibilities of each member of the family. If we fail to understand and call for this whole system of laws, then we will certainly fall in injustices, which the Shari`ah is not responsible for. It is our partial views and biased applications that are responsible for.
Thus, the components of this system, if you wish, are the following:
    1.    The laws of family support.
    2.    The laws of marriage.
    3.    The laws of wills.
    4.    The laws of inheritance.
 
1. The laws of family support state that a husband should support for his wife and the whole family. She is not financially responsible for that (except if people agree otherwise according to customs, etc). The related verse is stating:
(Men (Husbands) shall support and take full care of women (wives) with the bounties which God has bestowed abundantly …) (An-Nisaa’ 4:34)
Moreover, the same man, who is a husband and a father, is responsible for his parents, if they need his financial help, as well as his ‘near of kin’, i.e., sisters, aunts, etc. Legally, according to the Islamic law, a parent or a sister could ask for support from the man even through court. The Qur'an states, for example:
(And worship God [alone], and do not ascribe divinity, in any way, to aught beside Him. And take good care of your parents, and near of kin, and unto orphans, and the needy,) (An-Nisaa’ 4:36)
 
2. Regarding the laws of marriage, men have an additional financial responsibility to offer a gift (dowry) to the bride, the amount of which is also subject to customs.
(And give unto women their marriage portions in the spirit of a gift; but if they, of their own accord, give up unto you aught thereof, then enjoy it with pleasure and good cheer.) (An-Nisaa’ 4:4)
 
3. Before dividing the inheritance, a will/bequest should be fulfilled. Every person (man or woman) has the right to write such will. In fact, the Qur'an states within the ‘verses of inheritance’ (An-Nisaa’ 4:11-12):
(It is ordained for you, when death approaches any of you and he is leaving behind much wealth, to make a will (bequests) in favour of his parents and [other] near of kin in accordance with what is fair: I this is binding on all who are conscious of God. (2:180) And if anyone alters such a provision._ after having come to know it, the sin of acting thus shall fall only upon those who have altered it. Verily, God is all-hearing, all-knowing.) (Al-Baqarah 2:180-181)
Some scholars (notably Imam Ash-Shafie) narrated a hadith ahad (i.e., with a single chain of narrators) in which the Prophet was supposed to have said: ‘No bequest to an inheritor’, and they make this hadith an ‘abrogating’ (nasikh) to the verse. Without getting into too much theoretical analysis, it is a well-known fact, even according to Imam Ashl-Shafie himself- that such hadith (however strong it is) cannot cancel/abrogate a verse. Thus, most scholars agree to the principle of fulfilling the will of the deceased, as long as it is ‘reasonable’. The Qur'an explains how a will could be reasonable:
(… after [the deduction of] any bequest/will that may have been made, or any debt [that may have been incurred], neither of which having been intended to harm [the heirs] …) (An-Nisaa’ 4:12)

So, as long as the will is fair and is not intended to harm any other heir, it should be fulfilled.
Now, if the heirs differ over the will and its terms, then they should reach a settlement according to what is fair! The Qur'an states:
(If, however, one has reason to fear that the testator has committed a mistake or a [deliberate] wrong, and thereupon brings about a settlement between the heirs, he will incur no sin [thereby]. Verily, God is much-forgiving, a dispenser of grace.) (2:182)
 
4. Now comes the laws of inheritance, after the fact that men support for their families, and their extended families too, that men pay dowries, and that wills or bequests are supposed to be fulfilled within certain ‘reasonable limits’ that people agree on. What is left after the will (and paying the deceased’s debts as well) should be divided as follows:
(Men shall have a share in what parents and kinsfolk leave behind, and women shall have a share in what parents and kinsfolk leave behind, whether it be little or much – a share ordained [by God]…. ) (An-Nisaa’ 4:7)
(Concerning [the inheritance of] your children, God enjoins [this] upon you: The male shall have the equal of two females’ share; but if there are more than two females, they shall have two-thirds of what [their parents] leave behind; and if there is only one daughter, she shall have one-half thereof. And as for the parents [of the deceased], each of them shall have one-sixth of what he leaves behind, in the event of his having [left] a child; but if he has left no child and his parents are his [only] heirs, then his mother shall have one-third; and if he has brothers and sisters, then his mother shall have one-sixth after [the deduction of] any bequest he may have made, or any debt [he may have incurred]. As for your parents and your children – you know not which of them is more deserving of benefit from you: [therefore this] ordinance from God. Verily, God is all-knowing, wise. (4:12) And you shall inherit one-half of what your wives leave behind, provided they have left no child; but if they have left a child, then you shall have one-quarter of what they leave behind, after [the deduction of] any bequest they may have made, or any debt [they may have incurred]. And your widows shall have one-quarter of what you leave behind, provided you have left no child; but if you have left a child, then they shall have one-eighth of what you leave behind, after [the deduction of] any bequest you may have made, or any debt [you may have incurred]. And if a man or a woman has no heir in the direct line, but has a brother or a sister, then each of these two shall inherit one-sixth; but if there are more than two, then they shall share in one-third [of the inheritance], after [the deduction of] any bequest that may have been made, or any debt [that may have been incurred], neither of which having been intended to harm [the heirs]. [This is] an injunction from God: and God is all-knowing, forbearing. (4:13) These are the bounds set by God. And whoever pays heed unto God and His Apostle, him will He bring into gardens through which running waters flow, therein to abide: and this is a triumph supreme. (4:14) And whoever rebels against God and His Apostle and transgresses His bounds, him will He commit unto fire, therein to abide; and shameful suffering awaits him.) (An-Nisaa’ 4:11-14)
The above is the ‘default case’, if you wish. And yes, a brother could inherit more than his sister, but her husband is financially responsible for her if she is married, her father is financially responsible for her if she is single, and her brother himself is financially responsible for her if she ever needed money. Also, the brother pays dowry while his sister does not, etc. And before all that, if their father feels that his daughter needs a larger share in his inheritance, he could give her more than her share according to his will, which, again, should be written in reasonable terms, otherwise, the brother, for example, has the right to protest it. And so on.

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Source: www.onislam.net — Ask about Islam — Jasser Auda.