It seems that the proof the ring existed is rather weak. Granted, we have few materials outside the Bible about the Israelites dating from 1000-1 B.C., so it's not completely abnormal that a ring would not be mentioned in sources from that time. Still, that's a long gap that we have with no mention of it.
Supposedly the ring was used to build the Temple. But was that really necessary? The Bible says thousands of heathens were used and Lebanon's king helped. So why must we propose that the Bible's writers were thinking about a magical ring? Unclean spirits really aren't described as builders of people's cities from Israelite times in the Old Testament AFAIK.
Granted, the base of the Temple Mount is quite curious, and it deserves a separate thread. Suffice it to say that one could make a legend about the building based on how mind boggling it is. after Herod enlarged it, it was 5 times the size of the Acropolis, and had huge stones placed next to eachother with no gaps or cement. Those base rocks remind me a bit of the curious big block walls from the Andes. The stones around the corners are 100-150 tons, and there is a big stone inside the western wall that is 570 tons, the biggest one lifted in world history without powered machinery:

Naturally, the "ancient aliens" thinkers propose that the base of the Mount was built long before Solomon, while another hypothesis is that Solomon used some ingenious or very unusual technology to build it. It is similar to the surprising huge artificial base found at Baalbek in Lebanon, and this by the way can remind you that it was Lebanon's king who supposedly helped Solomon.
The second piece of the puzzle we have is the issue of demons from David's and Solomon's time. Despite some commentators' claims that the issue of demons was nonexistent before the Babylonian captivity, we read several mentions of unclean spirits from before that time. For example, it's said that Saul had an unclean spirit that endangered David.
It's not a complete surprise then that the Psalms might have something to do with that topic. In the Qumran caves' scrolls there were 4 Psalms in a row dealing with demons, leading up to and including the Bible's Psalm 91, which talks about God's protection from dark forces. In the Greek Septuagint, Psalm 91 is ascribed to David, and in the Targums it is ascribed to him and Solomon.
The second of those four Psalms (the first three aren't in the Bible) is connected with Solomon. It doesn't talk about Solomon using the demons, but rather protection from demons. While the Talmud said Solomon's ring had God's name on it, you may notice that this second Psalm talks about God's name. It repeats it ("Yahweh"), which is unusual I think for the Old Testament. For example, this Psalm says:
- healing [...] leans upon your name, and calls [...] He says to Israel, "Hold fast to YHVH, ... who made the heavens and the earth, and all that is in them, who separated light from darkness.
The full text is at:
judahgabriel.blogspot.com/2007/02/dead-sea-scroll-exorcism-psalms.html
However, while
Wisdom of Solomon 7 talks about his knowledge of spirits, next to other things it's useful to know about, in this Psalm it is much more about protection than about using them.