Introduction

A Ring is a set together with two binary operations + and . satisfying various axioms.

The "prototype" example is the set of integers Z with usual arithmetic. The fact that this example on its own gives the whole of "Number Theory" shows what a rich structure rings can have.

In fact, many of the "usual" examples where one can "add" or "multiply" give us rings. For example: Q, R, C, real valued functions, ...

However, starting with the axioms and looking for examples of things that satisfied them is not the way rings first came into mathematics.

Reference:

R B J T Allenby, Rings, Fields and Groups, 1991 [Now out of print, but on reserve in library].