Why Read This Book? - Index of

296 Chapter 9 Rings
9.3 Ring Properties
Since a ring is an abelian group under its addition operation, all properties
of abelian groups you proved in Chapter 8 apply to addition. With regard to
multiplication and its interaction with addition, it would probably be a good idea
to swing back through Chapters 2 and 3 and point out theorems that we proved for
the real numbers that exploited only their ring properties. A lot of theorems will
then translate directly over to a general ring. The only difference is that multiplication
might not be commutative, so we have to state and prove certain theorems
in two-sided language to get the full strength. We will state the theorems here, with
appropriate comments along the way. You will prove some of them as exercises.
The corollaries should be mere observations.
EXERCISE 9.3.1 If R is a ring, then a · 0 = 0 · a = 0 for all a ∈ R.
Exercise 9.3.1 implies that zero will not have a multiplicative inverse in a ring with
unity.
EXERCISE 9.3.2 If R is a ring and a, b ∈ R, then
(a) (−a)b =−(ab)
(b) a(−b) =−(ab)
(c) (−a)(−b) = ab
Corollary 9.3.3 If R is a ring with unity e, then (−e)a = a(−e) =−a for all a ∈ R.
In an abelian group with operation ∗, (a ∗ b) −1 = a −1 ∗ b −1 . Since a ring is an
abelian group under addition, this translates to the following additive form.
Theorem 9.3.4 If R is a ring, then −(a + b) = (−a) + (−b) for all a, b ∈ R.
The distributive property extends nicely in a general ring to yield the following
result analogous to Exercise 3.4.15 and Theorem 3.4.16.
Theorem 9.3.5 If R is a ring and a, b1,b2,...bn ∈ R, then
a � n k=1 bk = � n k=1 (abk) (9.15)
Theorem 9.3.6 If R is a ring and a1,a2,...,am,b1,b2,...bn ∈ R, then
��mj=1 ���nk=1 � �mj=1��nk=1 �
aj bk = ajbk
(9.16)
As in a group, if a ring has a unity element, it can have only one. Your proof
from Exercise 8.1.23 would translate directly over to a general ring, pretty much
word for word.