Hard, non-porous gemstones can be safely cleaned in warm water and a mild detergent. Use a small, soft brush to remove any build-up around the stone, taking care not to disturb or loosen the setting.
Rinse thoroughly after cleaning and dry/buff with a soft cotton cloth, such as would be suitable for cleaning gold jewellery. Store carefully, to avoid causing scratching to other, softer stones/materials you may have in your jewellery box, as much as for their own protection.
Diamond, sapphire, ruby, topaz, garnet, cubic zirconia and aquamarine are all examples of hard, non-porous gemstones.