Why Read This Book? - Index of

294 Chapter 9 Rings
A subring S of a ring R is merely a subgroup of the additive group that is also
closed under multiplication. We call {0} the trivial subring, and all subrings other
than {0} and R itself are called proper subrings.
Example 9.2.1 The integers are a subring of the rationals, and the rationals are
a subring of the real numbers. �
If R has a unity, it is not necessary that the unity be in S in order for S to be a
subring of R.
Example 9.2.2 Let m be a positive integer, and write mZ ={km : k ∈ Z}, the
set of integer multiples of m. This is another common notation for the set in
Eq. (8.31), where we denoted by (m) the subgroup of the additive group of integers
generated by m. Then mZ is a subring of the integers, because it is a subgroup of
the integers under addition, and it is closed under multiplication. If m ≥ 2, then
mZ does not contain 1. �
EXERCISE 9.2.3 Find all subrings of Z6 and Z7.
Example 9.2.4 Z2×2 is a subring of R2×2. �
EXERCISE 9.2.5 Call a square matrix diagonal if its only nonzero entries lie
on the main diagonal. Let D2×2 be the subset of Z2×2 consisting of the diagonal
matrices. Then D2×2 is a subring of Z2×2.
Example 9.2.6 In this example, we create a subring of the rational numbers that
we will return to in Section 9.9. Let QOD be the subset of the rational numbers
whose denominators are odd. There is more than one way to denote an element
of QOD. The form
� �
m
: m, n ∈ Z
(9.12)
2n + 1
is an obvious way. But another useful way to denote the set is to exploit the prime
factorization of the numerator and denominator, isolating 2 to keep it separate
from all the other primes involved. This works for all elements except zero, which
we throw in separately.
QOD ={0}∪
�
± 2n p1p2 ···pr
q1q2 ···qs
: n ∈ W and pi,qi are odd primes
�
for all 1 ≤ i ≤ r and 1 ≤ i ≤ s
(9.13)
The amount of repetition among the pi and qi does not matter. And notice that
the form of an element in Eq. (9.13) includes 1 by letting n = 0, r = s = 1 and