The fourth woman of the “begets” is not even given a name but there is no doubt who she is; David the king begot Solomon by her who had been the wife of Uriah. Matt 1:6
This is Bathsheba.
Late one afternoon David got out of bed after taking a nap and went for a stroll on the roof of the palace. As he looked out over the city, he noticed a woman of unusual beauty taking a bath. He sent someone to find out who she was, and he was told, “She is Bathsheba, the daughter of Eliam and the wife of Uriah the Hittite.” Then David sent for her; and when she came to the palace, he slept with her. 2 Sam 11:2-4 NLT
Later on, she sends word that she is pregnant.
David tries to deny or cover up the fact that he is the father but ends up guilty of the murder of her husband.
When Bathsheba heard that her husband was dead, she mourned for him. When the period of mourning was over, David sent for her and brought her to the palace, and she became one of his wives. Then she gave birth to a son. But the LORD was very displeased with what David had done. 2 Sam 11:26-27 NLT
The prophet Nathan tells David about his sin and David repents; but the wages of sin is death. Rom 6:23 NLT
Nathan replied, “Yes, but the LORD has forgiven you, and you won’t die for this sin. But you have given the enemies of the LORD great opportunity to despise and blaspheme him, so your child will die.” 2 Sam 12:13-14 NLT
The child does die but God’s grace is shown in the birth of the next son by the same woman, Bathsheba.
That child’s name is Solomon, who became king, Wiseman, and architect and builder of the great Temple.
Bathsheba is mentioned as an ancestor of Jesus reminds us of the Promise of Forgiveness.
In Him we have redemption through His blood, the forgiveness of sins, according to the riches of His grace. Eph 1:7
I believe her name is not specifically mentioned because of David, a man after God’s own heart. His sins, in this case, would give others occasion to blaspheme the name of God.
Yet, she is as much to blame for the adultery as he is; but she’s not responsible for the lying, deceit, and murderous plans.
Still, both find forgiveness in the lineage of the Savior.