As I used to ask my students all the time... "Did you really read the book, or just watch the movie?"
Gandalf speaking, in the chapter Shadow of the Past, Book One.
In Eregion long ago many Elven-rings were made, magic rings as you call them, and there were, of course, of various kinds: some more potent and some less. The lesser rings were only essays in the craft before it was full-grown, and Elven-smiths they were but trifles - yet still to my mind still dangerous to mortals. But the Great Rings, the Rings of Power, they were perilous.
A few pages later.... Still Gandalf....
'No,' said Gandalf, 'but I can. The letters are Elvish of an ancient mode, but the language is that of Mordor....'
So, if you have the tattoo of the script as it appears in the book, what you have is not Mordor - I don't think a written version of Mordor is found anywhere in the book, but Elvish.
Sauron helped the Elves to make the ring, but the real ring magic is the Elf's bad act, and as a result one of the outcomes is that they are pretty much self-banished from Middle Earth.