Why Read This Book? - Index of

316 Chapter 9 Rings
existence of multiplicative inverses. Thus there is only one thing to show in the
next exercise.
EXERCISE 9.8.11 A finite integral domain is a field. 6
By Exercise 9.8.11, we have that Zp is a field if and only if p is prime.
EXERCISE 9.8.12 Find multiplicative inverses for all nonzero elements of Z7.
In the integers, you showed in Exercise 2.5.5 that if a | b and b | a, then a =±b.
Furthermore, ±1 are the units in the integers. In a general domain, we say that
a and b are associates if a | b and b | a. Since this definition does not apply to
zero, we declare zero to be an associate of itself. Notice that this declaration
and the definition of associate prevent zero from having any other associates in
a domain.
EXERCISE 9.8.13 Show that 1 + i and 1 − i are associates in the Gaussian
integers Z[i].
EXERCISE 9.8.14 In a domain, two elements a and b are associates if and only
if there exist units u and v such that a = ub and b = va.
You should feel an equivalence relation coming on about now.
EXERCISE 9.8.15 Let D be a domain, and define a ∼ b if a is an associate of b.
Then ∼ is an equivalence relation on D. 7
Anytime you create an equivalence relation, it is natural to ask two questions:
What do the equivalence classes look like, and which element from each
equivalence class might be a good choice as a representative element of the class?
EXERCISE 9.8.16 If D is a domain and ∼ is the equivalence relation of
association, then [e] is the set of units of D.
If a and b are associates, then there ought to be some senses in which they
are interchangeable. One example of how this is true is in the integers, where
(6) = (−6). Associates generate the same principal ideal. The best way to show
this is first to prove the following. Its corollary is immediate.
EXERCISE 9.8.17 Suppose a and b are elements of a domain. Then a | b if and
only if (b) ⊆ (a).
6 To find a −1 , define f(x) = ax and apply Exercise 4.6.11.
7 Don’t forget zero.