I own an online jewelry business and always err on the side of caution. Ivory soap or any plain soap is best, however I like to steam my jewelry. This cleans metals like tarnished silver very well and I know that many jewelers prefer this method. Just boil some water in a tea kettle and wait until it starts to steam. Then, using tongs hold your ring in the steam for a few minutes being careful not to burn yourself. This is a great disinfectant as well. Double-check if you can do this with peridot, but I clean my diamonds that way.
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