Kommentar zu Bereschit 21:10

עם בני עם יצחק WITH MY SON, WITH ISAAC —For since this is my son, even though he were not as good as Isaac really is, or if he were as good a man as Isaac really is, even though he were not my son, this one (Ishmael) is not deserving of inheriting with him: how much less is he deserving of inheriting with my son, with Isaac — with one who possesses both these qualities (of being my son and of being the good man Isaac) (Genesis Rabbah 53:11).

For the son of this maidservant shall not inherit: And [just] because he banished him, is it impossible for him to return for his inheritance after the death of his father? So it appears that it was speaking about the inheritance of his deeds. As Sarah saw the son of Hagar the Egyptian, whom she bore for Avraham, laughing with sexual immorality - from the angle that he was the son of Hagar the Egyptian, [as] she gave birth to one similar to herself. For it is known that the Egyptians are awash in licentiousness, as it is stated (Ezekiel 23:20), "and whose issue was like that of horses"; and their offspring are similar to them. Therefore Sarah said to banish him, so that Yitzchak would not learn from his deeds. But perhaps you will say, it is the opposite - maybe Yitzchak will change him and bring him back to the good! It is about this that she said, "I knew it through prophecy that he will not inherit with my son from the good deeds of his father, Avraham." And hence the concern remained that Yitzchak would learn from him. And [for] more of the content of this section, see below, Parashat Chaye Sarah, on the verse, "and her name was Keturah" (Kli Yakar on Genesis 25:1).

גרש האמה הזאת ואת בנה, seeing that what the son did was at the instigation of his mother. He spread lies about Yitzchok’s legitimacy in order to establish a claim to your inheritance.