"The One Ring" is almost certainly a reference to the Rings of Power in Tolkien's Lord of the Rings universe: three for elves, seven for dwarves, nine for men, and one to rule them all.
This last one was often referred to as "the One Ring".
Az egy gyűrű. as a normal sentence should also mean "That is a ring."
Note, though, that Duo's example does not have a full stop (period) at the end, indicating that this is not a sentence, making the "noun phrase" interpretation more likely than the "sentence" one.
The words are capitalized because it's a title "The One Ring" from the book and movie "The Lord of the Rings".
This is a tough one because egy means both "a" and "one", and az means both "the" and "that". Normally when the meaning is "that" there's still an "a/an" or "the" before the noun. Az az asztal fehér (that table is white).
I am a student like you so if any native speakers correct what I said, believe them!