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Proofs in Ring Theory
Chapter 14
Proofs in Ring Theory
We have noted that many of the proofs we have seen thus far involve integers and their properties.
This was certainly the case in Chapter 11, where we were primarily concerned with additive and
multiplicative properties of integers. Many important properties of integers follow from just a
very few familiar additive and multiplicative properties of integers. In particular, every three
integers a, b, and c satisfy the following:
(1)
a + b = b + a
(2)(a + b) + c = a + (b + c)
(3)
a + 0 = a
(4)
a + (
a) = 0
(5)
a(bc) = (ab)c
(6)
a(b + c) = ab + ac
(14.1)
Properties (1) (4)tell us that the integers form an abelian group under addition, a fact we
observed in Chapter 13. You can probably think of other familiar properties of integers (such
as ab = ba), but lets concentrate on the six properties listed above. We saw in Chapter 13
that some of these properties have names. For example, (1)is called the commutative law of
addition; while (2)and (5)are the associative laws of addition and multiplication, respectively.
Property (6)is called the distributive law. Property (3)states that the integer 0 is the identity
under addition; while property (4)tells us that for an integer a, the integer
a is its inverse
under addition. Properties (3)and (4)in particular may seem as if they are such basic properties
of the integers that they should not even be mentioned. However, it is precisely that these six
properties are so basic and natural that makes them important and draws our attention to them.
A question now arises: Just what facts about the integers are consequences only of these
six properties? An even more basic question is: If we have a nonempty set S of objects (not
necessarily integers)for which it is possible to add and multiply every two elements of S (and
in each case obtain an element of S)such that properties (1)- (6)are satised, then what
additional properties must S possess? Of course, whatever properties that can be deduced
about the elements of S will be properties of the integers as well.
In fact, this is the essence of the area of abstract algebra that we are about to encounter
(and often of all mathematics). While studying a familiar set of objects, we may discover an
interesting fact about this set. But what features of this set led us to this conclusion? And
if any other set had these same features, does this interesting fact hold for these sets as well?
We are now prepared to explore nonempty sets on which addition and multiplication have been
dened that satisfy properties (1)- (6).
14.1 Rings
Addition and multiplication of integers are binary operations since each associates an integer
with each (ordered)pair of integers. Binary operations in general are discussed in Chapter 13.
1 2
CHAPTER 14. PROOFS IN RING THEORY
In the current context, a nonempty set with one or more binary operations that are required
to satisfy certain prescribed properties is referred to as an algebraic structure. Hence we
have already seen examples of algebraic structures. Indeed, every group (see Chapter 13)is an
algebraic structure. Studying algebraic structures is fundamental to abstract algebra.
We mentioned that the familiar operations of addition and multiplication dened on the
integers satisfy the six properties listed in (14.1). Other familiar sets of numbers with these
operations also satisfy these six properties, including the rational numbers, the real numbers,
and the complex numbers. The situation is dierent for the irrational numbers, however, since
addition and multiplication are not even binary operations. For example, 2 and
2 are
irrational numbers while 2
· 2 = 2 and 2 + (2)= 0 are not.
These and other examples suggest a general concept. A set R (this is not the symbol used
for the set of real numbers)with two binary operations, one of which is called addition and
denoted by + and the other called multiplication and denoted by
· (where we often write ab
rather than a
· b for a, b R), is called a ring if it satises the following six properties:
R1 Commutative Law of Addition: a + b = b + a for all a, b
R;
R2 Associative Law of Addition: (a + b) + c = a + (b + c)for all a, b, c
R;
R3 Existence of Additive Identity: There exists an element 0
R such that a + 0 = a for
all a
R;
R4 Existence of Additive Inverse: For each a
R, there exists an element a R such
that a + (
a) = 0;
R5 Associative Law of Multiplication: a(bc) = (ab)c for all a, b, c
R;
R6 Distributive Laws: a(b + c) = ab + ac and (a + b)c = ac + bc for all a, b, c
R.
Notice that property R3 requires the existence of at least one element in R, which implies
that every ring is nonempty. Recall that if S is a set with a binary operation
and e S
is an identity for S under
, then e a = a e = a for all a S. Since a ring R has two
binary operations and an identity element is only required for the operation of addition, we
refer to an element 0 specied in property R3 as an additive identity. The notation 0 for an
additive identity is chosen because the integer 0 is an additive identity in Z. In other words,
an additive identity in a ring R has the same characteristic as the integer 0 under addition in
Z. It is important to realize that when we refer to an additive identity 0 in a ring R, we are
referring only to an element in R that we are denoting by 0 and that satises property R3,
namely, a + 0 = a for all a
R. Since property R1 holds in every ring, we also have 0 + a = a.
Also, if an algebraic structure (S,
)has an identity e, then an element a S has an inverse
b
S if a b = b a = e. Each element of a ring R is only required to have this property for the
operation of addition. Thus, an inverse of an element a
R with respect to addition is called
an additive inverse of a. In Z, an additive inverse of an integer m is its negative
m. For
this reason, we use
a to denote an additive inverse of an element a in a ring R. We must keep
in mind that an element
a in R stands only for some element in R that satises property R4,
namely, a + (
a)= 0. By property R1, we also know that (a) + a = 0. Since properties R1
R4 are required of every ring R, it follows that (R, +)is an abelian group.
A ring with binary operations + and
· is commonly denoted by (R, +, ·). However, if the
two operations involved are clear, then we simply write R. In particular, if we are dealing
with a familiar set with standard operations of addition and multiplication (and these are the 3
operations we are using), then we write only the symbol for that set. Thus Z, Q, R, and C are
rings.
We now look at some other common examples of rings.
Result 14.1
The set 2Z of even integers is a ring under ordinary addition and multiplication.
Proof.
First we show that ordinary addition and multiplication are binary operations on 2Z.
Let a, b
2Z. Then a = 2x and b = 2y for x, y Z. Then a + b = 2x + 2y = 2(x + y)and
ab = (2x)(2y)= 2(2xy). Since x + y and 2xy are integers, a + b and ab belong to 2Z.
Since 2Z
Z and the binary operations in 2Z are the same as those in Z, properties R1,
R2, R5, and R6 are automatically satised. Moreover, since the integer 0 is even, 0
2Z and so
2Z has an additive identity. To show that property R4 is also satised, let a
2Z. So a = 2x,
where x
Z. Then a = (2x) = 2(x). Since x Z, it follows that a 2Z.
Result 14.2
The set Z
n
=
{[0], [1], [2], · · · , [n 1]}, n 2, of residue classes is a ring under
residue class addition and residue class multiplication.
Proof.
It was indicated in Chapter 7 that both residue class addition and multiplication
dened by [a] + [b] = [a + b] and [a]
· [b] = [ab] are well-dened and so are binary operations
in Z
n
. That properties R1, R2, R5, and R6 are satised depends only on the corresponding
properties in the ring Z. For example, to see that R1 and R2 are satised, let [a], [b], [c]
Z
n
.
Then
[a] + [b] = [a + b] = [b + a] = [b] + [a]
and
([a] + [b])+ [c]
=
[a + b] + [c] = [(a + b) + c] = [a + (b + c)]
=
[a] + [b + c] = [a] + ([b] + [c]).
The proofs of properties R5 and R6 are similar. The residue class [0] is an additive identity in
Z
n
and an additive inverse for [a] is [
a] since [a] + [a] = [a + (a)] = [0].
The ring (Z
n
, +,
·)described in Result 14.2 is commonly called the ring of residue classes
modulo n.
Result 14.3
The set M
2
(R) of 2
× 2 matrices over R is a ring under matrix addition and
matrix multiplication.
Proof.
Recall that for A =
a
b
c
d
and B =
e
f
g
h
in M
2
(R), addition and multiplication
are dened by
A + B =
a + e
b + f
c + g
d + h
and AB =
ae + bg
af + bh
ce + dg
cf + dh
.
An additive identity for M
2
(R)is the zero matrix Z =
0
0
0
0
and an additive inverse for the
matrix A given above is the matrix
A = a b
c d . The verication of properties R1, R2,
R5, and R6 depends only on the properties of the ring R.
Not only is M
2
(R)a ring under matrix addition and matrix multiplication, so too is M
n
(R)
for each integer n
2. 4
CHAPTER 14. PROOFS IN RING THEORY
Result 14.4
The set
F
R
=
{f : f : R R} of real-valued functions with domain R is a
ring under function addition and function multiplication.
Proof.
Recall that for f, g
F
R
, addition and multiplication are dened by
(f + g)(x) = f (x) + g(x)and (f
· g)(x) = f(x) · g(x)
for all x
R. The proofs of properties R1, R2, R5, and R6 depend only on properties of the
ring R. For example, property R1 follows because
(f + g)(x) = f (x) + g(x) = g(x) + f (x) = (g + f )(x)
for all x
R and so f + g = g + f; while property R5 follows because
((f
· g) · h)(x) = (f · g)(x) · h(x) = (f(x) · g(