Sforno on Genesis 21:10/ספורנו על בראשית כ״א:י׳

1גרש 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.

2כי לא יירש בן האמה, seeing that he is not part of your family; the child always is considered as part of the halachically defective parent. (Maimonides issurey bi-ah 15,7)