Bear in mind that the Talmud says that King Solomon never actually worshipped any idols, but since he did not stop his wives from doing so when he could have, Hashem considered it as if it were King Solomon's own sin.)
Muman, I don't know where in Talmud it is stated that King Solomon never actually worshiped any idols. Since I have not done enough research about Talmud, I am unable to pull up the corresponding verse(s)
The following verses from the Tanach unambiguously prove that King Solomon indeed worshiped idols...
1 Kings 11:4
As Solomon grew old, his wives turned his heart after other gods, and his heart was not fully devoted to the LORD his G-d, as the heart of David his father had been.
1 Kings 11:5
He followed Ashtoreth the goddess of the Sidonians,
and Molech the detestable G-d of the Ammonites.
1 Kings 11:6
So Solomon did evil in the eyes of the LORD; he did not follow the LORD completely, as David his father had done.
1 Kings 11:7
On a hill east of Jerusalem, Solomon built a high place for Chemosh the detestable G-d of Moab,
and for Molech the detestable G-d of the Ammonites.
1 Kings 11:33
I (HaShem) will do this because they have forsaken me and
worshiped Ashtoreth the goddess of the Sidonians, Chemosh the G-d of the Moabites, and Molech the G-d of the Ammonites, and have not walked in my ways, nor done what is right in my eyes, nor kept my statutes and laws as David, Solomon's father, did.
Even if we assume that Solomon did not directly worship the idols the details about the wife of Solomon and the queen mother of Solomon's successor raises questions!
Also Solomon's wife and the mother of Solomon's successor King Rehoboam was an Ammonite woman called Naamah... Since Solomon worshiped the detestable goddess of Ammon and built alter for that idol I would assume Naamah played a role in leading Solomon astray! Solomon gave importance to Naamah, the Ammonite queen which resulted in her son becoming Solomon's successor. Naamah never converted to Judaism so King Rehoboam was also not Jewish!!! Again imagine a non-Jewish king succeeding Solomon!!!
It would also mean King David's grandson who ascended to his throne was NOT Jewish?!?! Unbelievable! 2 Chronicles 12:13 (also in 1 Kings 14:21)
King Rehoboam established himself firmly in Jerusalem and continued as king. He was forty-one years old when he became king, and he reigned seventeen years in Jerusalem, the city the LORD had chosen out of all the tribes of Israel in which to put his Name. His mother's name was Naamah; she was an Ammonite.
Surprisingly Namaah was the only one of Solomon's wives to be mentioned, within the Tanakh, as having borne a child.