“And Avraham arose up early in the morning, and took bread and a bottle of water, and gave it unto Hagar, putting it on her shoulder, and the child, and sent her away….” B’rashith 21.14
Yishmael (Ishmael) forfeited the right to be an heir when he mocked the covenant of circumcision; as it says, “And Sarah saw Ishmael mocking” which our Sages state that “mocking” means, “the laughter of derision” – In relationship to B’rashith 21.10 and 22.1 and “And it came to pass after these words” (Genesis 22.1) hade’varim :
And some say, “ after the words of Ishmael,” who was boasting to Isaac that he was circumcised at the age of thirteen, and he did not protest. Isaac said to him,“ With one organ you intimidate me? If the Holy One, blessed be He, said to me, ‘Sacrifice yourself before Me,’ I would not hold back.” – Cf. Gen. Rabbah 55:4.
Ishmael never had or possessed “the birthright” as he was only a “potential heir” as it says, “that ‘I’ may be built up from her” (Gen. 16.2); and, “cast out the bondwoman and her son, for the son of this handmaid shall not inherit with my son, with Isaac.”” Gen. 21.10
From this you learn that “with Isaac” can be read in one of two ways – as it says, “shall not inherit with Isaac” – shall not inherit with laughter (mockery). “With Isaac” seems to be superfluous and can be read two ways, “with my son Yitzchaq” or “with laughter/mockery” –
That is, it can be read as, Yishmael shall not inherit [together] with Yitzchaq; or it can be read, “with laughter/mockery” (e.g. Judges 16:25) as the passuk (Tehillim 30.2) says, “I will extol thee, O L-RD, for Thou hast raised me up, and hast not suffered mine enemies to rejoice over me.”
As it is related, Melek (King) David prophesied in Tehillim 105.9 [The covenant] which He made with Abraham, and His oath unto Isaac; לְיִשְׂחָק
“Netsiv goes on to point out that Psalms 105, which is identical to chapter 16 in Chronicles I in all but one verse, foretells a better future, in which also Yitshak’s name will be altered. Referring to God’s covenant with the patriarchs, the ninth verse in this Psalm differs from its correlate (Chronicles I, 16:16) in that the name Yitshak is spelled Yishak in the phrase: “Ushevua-to leYitshak.” Says Netsiv, David foretold through divine inspiration that some day, the patriarch would be renamed to indicate the full joy that would replace the mockery.” Rabbi Naftali Zvi Yehuda Berlin of Volozhin, Ha’amek Davar: Commentary on the Pentateuch, New York: Friedman.
You see from the prophetic words of Sarah, that Avraham manumitted Hagar where-in under the Code of Lipit-Ishtar Hagar is set free: the Code of Lipit-Ishtar it stipulates that “if a father grants freedom to a slave-woman and the children she has borne him they forfeit a share of the paternal property” (Judges 11.1-3).