Morgoth did have a ring once, which he planned to give to Ungoliant. However, they had big fight after that Two Trees incident.
"You mean you've got three Silmarils, and you're giving me this lousy cubic zirconia?!"
It all went downhill after that.
If Morgoth did have a ring, it might have been decorative or purely symbolic (like a signet ring) rather than having any power. Morgoth's Ring was all of Arda after he corrupted it at its making.
"Just as Sauron concentrated his power in the One Ring, Morgoth dispersed his power into the very matter of Arda, thus the whole of Middle-earth was Morgoth's Ring." - J.R.R. Tolkien