Ah... another brilliant "feature" of iTunes 11! There are too many stupidities in this new version to even attempt to itemize here, but this one is truly amazing. I ran into this problem after buying some classical music from the iTunes Store. The tracks had the composer's name in the "artist" field and nothing in the "composer" field. Changing the contents of "artist" seems to work, as icsbicr noted, but then if you play the track, or do Cmd-I, or various other things, the artist changes back to the original (and, in this case, incorrect) content. If someone at Apple thinks we ought not to be allowed to change "artist" for purchased content, then at least they ought to prevent us from doing it right away instead of seeming to change it and then later changing it back.
So here's how to fix it. You don't want to mess with Mickey Mouse circumventions like "various artists" and "part of a compilation" (as suggested in https://discussions.apple.com/thread/4607365?start=0&tstart=0), because those tags are INCORRECT. If you want the information to be correct, you've got to change the tags in the music file itself. I realize this is a huge pain, but Apple doesn't seem to care too much about causing their customers big pains. I used a product called Fission from Rogue Amoeba, though there are probably plenty of other products which will do the same thing.
You need to be careful so you don't get the iTunes data base out of sync with the file system structure. What I did was copy (not move) the files from the iTunes folders (use "Show in Finder" from the context menu if you're not sure where they are) to another folder somewhere and then deleted them from iTunes. Then I used Fission to edit the tags in the file to show the correct information and saved the files to another folder (keeping the original files as they were, just in case of trouble). Then I added the files with the edited tags to iTunes. The only small problem I can see is that the tracks no longer show up in the "Purchased" list, but I can live with that.
Good luck.