The movie diverges quite significantly from the book in this area. The movie script of the first (edited) memory contains this dialog:
Tom Riddle: I was in the library the other night. In the restricted section. And I picked something rather odd about this rare magic. I thought perhaps, you could illuminate me. It's called, as I understand it, Hor...
Horace Slughorn: I beg your pardon? I don't know anything about such things and if I did, I wouldn't tell you. Now get out of here at once and don't let me ever catch you mentioning them again!
So its clear from the script that in the tampered memory that Dumbledore originally has, Riddle doesn't mention Horcruxes explicitly (though the 'Hor...' surely would give Dumbledore a clue).
So I believe the script writers were trying to set up the situation where Harry is used to discover both the name and the magic used and reveal the extent that it was used.
I mentioned all that because I think it makes your question about the ring relevant. If Dumbledore discovered and destroyed a Horcrux already, why does he need the memory?
It is possible that Dumbledore only knows for sure that the ring contained a powerful curse. He may have suspected that it was a Horcrux, but needed the memory to confirm this. The real memory may also contain important additional information that they could use to defeat Voldemort, so he had additional good reasons to try to obtain it from Slughorn.