Why Read This Book? - Index of

9.4 Ring Extensions 301
in the integers, but n1 = � n k=1 1 = 0inZn. In a general ring with unity element e,
whether or not some expression ne = � n k=1 e = 0 ever occurs motivates a term.
Definition 9.3.24 Suppose R is a ring with unity, and suppose there exists a
positive integer n such that ne = 0. Then the smallest such n for which this holds
is called the characteristic of R, and is denoted char R. If no such positive integer
exists, then R is said to have characteristic zero.
In Zn the fact that n ≡n 0, or n1 is zero in Zn means that char Zn ≤ n. On the
other hand, if m is a positive integer and m ≡n 0, that is, if m1 is zero in Zn, then
m is a multiple of k, so that m ≥ n. Thus char Zn ≥ n, so that we have proved the
following theorem.
Theorem 9.3.25 If n ≥ 2, then char Zn = n.
EXERCISE 9.3.26 What is char(Z4 × Z18)? Explain.
If adding the unity element to itself n times produces a sum of zero, then the
same is true for all elements of the ring.
EXERCISE 9.3.27 Let R be a ring with unity and nonzero characteristic n. Then
nx = 0 for all x ∈ R.
9.4 Ring Extensions
We can create a very important type of algebraic structure from a given algebraic
structure by tossing in a new element, stirring well, and letting the mixture
expand into another algebraic structure of the same type. It is called the process of
adjoining an element in order to create what is called an extension of the original
structure. In this section we want to get acquainted with the creation of extensions
by adjoining elements to commutative rings. In principle, only two types of
ring extensions can result from adjoining an element. We will begin with a very
specific example of the first type, but instead of building it up as an extension of
a certain ring in the most rigorous way, we will just lay the whole structure out
there, define equality and the operations, and show that what we have presented
is a ring. But don’t worry. We will make up for our lax introduction of this ring
in Section 9.14, where we will see a more rigorous way to construct it. After we
have presented our example of the first type of ring extension, we will point out
how other extensions of the same type can be created through precisely the same
reasoning. Finally, we construct the canonical example of the second type. We will
use these constructions over and over throughout the rest of this chapter.
9.4.1 Adjoining Roots of Ring Elements
Example 9.4.1 Let S ={a + b 3√ 2 + c 3√ 4 : a, b, c ∈ Z}, the set of all integer linear
combinations of {1, 3√ 2, 3√ 4}. First, we define x = a1 + b1 3√ 2 + c1 3√ 4tobe