Rhodium does not tarnish, but it does wear off. Wedding and engagement rings plated in rhodium must go through frequent re-plating. However, most jewelry stores do not charge for this process if the ring was purchased at their establishment or another of their outlets.
All white gold jewelry is coated with a thin layer of rhodium. Bleach or pool chlorine causes rhodium to wear off faster. When the plating is new, the ring shines most brightly because of the whiteness of rhodium. Rhodium is one of the rarest metals, which is why white gold jewelry is generally more expensive than natural gold equivalents.