Generally with any jewelery, it's best to try not expose any jewelry to any chemicals, which include lotions and perfumes. Any stains and discolorations with jewelry are typically caused by environment exposure.
Although stainless steel is extremely durable, it can be damaged by harsh chemicals such as chlorine. Other metals such as gold, silver, platinum can be also be easily damaged by chlorine..
Chlorine affects stainless steel because it breaks down what is known as the ‘passive film’ on the surface of stainless steel. The passive film develops when stainless steel is exposed to certain types of compounds or solutions or through electropolishing. This film protects stainless steel from corrosion, but chlorine can literally eat through the passive film.
When chlorine eats through the passive film, pits develop on the surface of the stainless steel. And once this happens, the stainless steel is vulnerable to corrosion and further pitting and damage. Once the stainless steel is damage, it might be repairable by having a jeweler buff it.
There are several other ways to try to clean the stains off of stainless steel.
1 - Rub a pencil eraser over the stain as if you're erasing a mark on paper. Don't be afraid to rub hard -- the eraser will not damage the steel. Continue erasing until all or most of the stain comes off.
2 - Dip a soft cloth in white vinegar or club soda if the eraser fails to remove the stain. Gently rub the cloth over the stain until it comes off.
3 - cover the stain with dry baking soda, and then pour vinegar onto it, which will result in bubbling action. Gentle rub the stained area and see if the stain comes off.
4 - Fill a coffee cup three-quarters full with distilled water. Place the cup in the microwave and heat until the water is very warm but not hot. Squirt one pump of unscented, dye-free antibacterial soap into the warm water and stir with a clean spoon. Soak until the water turns cool. Pour distilled water over the jewelry to rinse it. Lay the jewelry on a clean paper towel to dry.
5 - Try a jeweler and see if the jewelry can be buff cleaned.
After cleaning, add a layer of protection with a layer of polymer such as products like "Jewelry Shield".