ABSTRACT
Preview Activity 8.1. Our study of rings began in Investigation 1, where we learned about the
integers and their various axioms. Our next example of a ring was Zn, a set of equivalence classes of integers. As you might suspect from these two examples, the integers play an important role in
the general theory of rings. In fact, even in rings whose elements are not integers, it is possible to
define notions of integer multiplication and integer exponentiation. In other words, it is possible to
multiply and exponentiate ring elements by integers, even though the ring elements themselves may
not be integers. In fact, it turns out that integer multiplication and exponentiation work exactly the
way we would expect them to. To see this, use your intuition to calculate as many of the quantities
listed below as you can. For those that you are not able to calculate, explain why. Throughout your
calculations, you will be applying the definitions that we will formally develop in this investigation.
