There's two. One for a no death run and one for a no bonfire run. Each ring will make either your left or right hand item invisible even with a buff.
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No bonfire run? That would take a lot of lifegems.
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No bonfire run is best done on ng+ so that you can be set with healing and repair items. I attempted this late on ng8. I got as far as black gulch..
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I'm guessing that's even harder than no death run, as if you die you end up at a bonfire. I don't even think it would be possible on a new character, think of how many weapons you would have to go through, or a long journey back to Majula.
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You can light bonfires but not sit at them. The only use able bonfires are the primordial ones. No bonfire run males it a long journey and needs careful planning but is easier than a no death run