In this section Hungerford defines a ring and gives several different examples. From them I gather that there are many different types of rings. But I think the type of rings that will be the most interesting shall be those where every axiom is met (see page 44).
When looking at a ring outside of the mathematical context we often see a piece of metal that has been completely sealed. There is no beginning nor end. Relating this back to math. Rings, from my current understanding (and Section 3.1 needs to be reviewed) a ring should be closed. It may be closed by the operations that may be preformed within a set, in one way or another there must be some sort of closure.
In physics we often study patterns, repetition, but more specifically conservation. A proper understanding of rings could reveal a lot about how the world works.