Yep, the stove rings get dirty pretty quickly. Who wants to rub them in perpetuity, surely not you!
The solution is simple: soak them overnight in a sealed plastic bag that you have poured ammonia into. The next day you will just have to rinse!
In addition, this solution is inexpensive since ammonia can be sold for just a few dollars, and it can be found everywhere where cleaning products are sold.
Note that it is not so much the liquid, but the ammonia vapors that will do the job, so no need to fill your entire plastic bag. Also, be sure to make your bag waterproof, as ammonia has a fairly strong odor.
BE CAREFUL: never mix bleach and ammonia.