Some think that Rahab is the harlot Rahab who received Joshua's spies: she saved them and saved herself. Matthew mentioned her in order to show that as she was a harlot, so was the whole assembly of the Gentiles, for they committed fornication in their deeds. But those of the Gentiles who received Jesus' spies, that is, the apostles, and believed in their words, were all saved.

Boaz begat Obed by Ruth.

This Ruth was a foreigner; nevertheless, she married Boaz. In the same way, the church of the Gentiles, being a foreigner and outside the covenants, forgot her people and the worship of idols, and her father the devil, and the Son of God took her to wife.

And Obed begat Jesse. Jesse begat King David, and King David begat Solomon by Uriah.

And Matthew mentions Uriah's wife here in order to show that one should not be ashamed of one's ancestors, but most of all try to glorify them also with one's virtue, and that all are pleasing to God, even if they come from a harlot, if only they have virtue.

Solomon begat Rehoboam. And Rehoboam begat Abijah. And Abijah begat Asa. Asa begat Jehoshaphat. And Jehoshaphat begat Jehoram. And Joram begat Uzziah. Uzziah begat Jotham. Jotham begat Ahaz. Ahaz begat Hezekiah. Hezekiah begat Manasseh. Manasseh begat Amun. Amon begat Josiah. And Josiah begat Jehoiakim. And Jehoiakim begat Jehoiachin and his brethren, before he migrated to Babylon.

The Babylonian migration is the name given to the captivity that was later endured by the Jews, who were all taken together to Babylon. The Babylonians also fought with them at other times, but embittered them more moderately, and at the same time completely removed them from their homeland.