Many of these people are giving you terrible advice. You should never, ever clean a Tanzanite stone with anything other than mild, soapy water. It is a delicate stone, and most of the methods people who responded have mentioned eventually will ruin your ring. Also, store your tanzanite ring separately from other jewelry, and do not expose it to extreme temperature changes.
Mix 1/2 ammonia and 1/2 hydrogen peroxide. Swish the ring around in this for about 30 seconds. You can see the dirt bubbling out.You'll want to clean all your jewelry when you see the results. Just don't use this formula for "soft" gems like pearls and opals.
If you have the money for some Tarnex it's great for everything from coins to silver to jewelry. But, toothpaste and a soft brush also work well in a pinch.
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