Why Read This Book? - Index of

300 Chapter 9 Rings
EXERCISE 9.3.20 Suppose R is a ring with unity element e. Show that
(me)(ne) = (mn)e for all positive integers m and n.
Even though multiplication in a ring is not necessarily accompanied by an
identity and inverses for elements, we can use the multiplicative forms of the
definitions of an in a limited way. For a ring element a, we begin by defining a1 = a
and an+1 = an · a for n ≥ 1. If R has a unity element e, we also define a0 = e, but
only for nonzero a.Ifa is a unit of R, we can define a−n = (a−1 ) n .
� �
1 2
Example 9.3.21 In Z2×2, let A = Then
−1 0
A 0 � �
1 0
= I =
0 1
A 1 � �
1 2
= A =
−1 0
A 2 � �
−1 2
= A × A =
−1 −2
A 3 = A 2 � �
−3 −2
× A = , etc. �
1 −2
EXERCISE 9.3.22 Evaluate the first few (multiplicative) powers of 3 and 5
in Z10.
With these definitions of a n for appropriate integers n, and by arguments
exactly like those in Exercise 3.5.4, we have the following.
Theorem 9.3.23 Suppose R is a ring, a, b ∈ R, and let m and n be positive
integers. Then
a m · a n = a m+n
(a m ) n = a mn
(9.29)
(9.30)
If R has a unity element and a is a nonzero ring element, Eqs. (9.29) and (9.30)
hold for all nonnegative integers m and n. Ifa is a unit, these equations hold for
all integers m and n. Furthermore, if R is commutative,
for all n for which a n and b n are defined.
(ab) n = a n b n
(9.31)
One big difference in the way we visualize the integers and Zn is that the
integers extend out indefinitely along the number line in both directions, whereas
Zn is circular. If we generate both of these rings by considering 1, 1 + 1, 1 + 1 + 1,
and so on, no expression of the form n1 = � n k=1 1 ever produces a sum of zero