Proof.
Exercise.
2
Example 5.1
The set
Z
under the usual rules of multiplication and addition forms a ring.
2
Example 5.2
For
n >
1, the set
Z
n
under the rules of multiplication and addition defined in
§
2.3
forms a ring. Note that
Z
n
with
n
= 1 does not satisfy our definition, since our definition requires
that in a ring
R
, 1
R
6
= 0
R
, and in particular,
R
must contain at least two elements. Actually, if we
have an algebraic structure that satisfies all the requirements of a ring except that 1
R
= 0
R
, then
it is easy to see that
R
consists of the single element 0
R
, where 0
R
+ 0
R
= 0
R
and 0
R
·
0
R
= 0
R
.
2
Example 5.3
The set
Q
of rational numbers under the usual rules of multiplication and addition
forms a ring.
2
Let
R
be a ring.
The
characteristic
of
R
is defined as the exponent of the underlying additive group. Alterna
tively, the characteristic if the least positive integer
m
such that
m
·
1
R
= 0
R
, if such an
m
exists,
and is zero otherwise.
For
a, b
∈
R
, we say that
b
divides
a
, written
b

a
, if there exists
c
∈
R
such that
a
=
bc
, in
which case we say that
b
is a
divisor
of
a
.
Note that parts 15 of Theorem 1.1 holds for an arbitrary ring.
5.1.1
Units and Fields
Let
R
be a ring. We call
u
∈
R
a
unit
if it has a multiplicative inverse, i.e., if
uu
0
= 1
R
for some
u
0
∈
R
. It is easy to see that the multiplicative inverse of
u
, if its exists, is unique, and we denote
it by
u

1
; also, for
a
∈
R
, we may write
a/u
to denote
au

1
. It is clear that a unit
u
divides every
a
∈
R
.
We denote the set of units
R
*
. It is easy to verify that the set
R
*
is closed under multiplication,
from which it follows that
R
*
is an abelian group, called the
multiplicative group of units
of
R
.
If
R
*
contains all nonzero elements of
R
, then
R
is called a
field
.
Example 5.4
The only units in the ring
Z
are
±
1. Hence,
Z
is not a field.
2
Example 5.5
For
n >
1, the units in
Z
n
are the residue classes [
a
mod
n
] with gcd(
a, n
) = 1. In
particular, if
n
is prime, all nonzero residue classes are units, and conversely, if
n
is composite,
some nonzero residue classes are not units. Hence,
Z
n
is a field if and only if
n
is prime.
2
Example 5.6
Every nonzero element of
Q
is a unit. Hence,
Q
is a field.
2
5.1.2
Zero divisors and Integral Domains
Let
R
be a ring. An element
a
∈
R
is called a
zero divisor
if
a
6
= 0 and there exists nonzero
b
∈
R
such that
ab
= 0
R
.
If
R
has no zero divisors, then it is called an
integral domain
.
Put another way,
R
is an
integral domain if and only if
ab
= 0
R
implies
a
= 0
R
or
b
= 0
R
for all
a, b
∈
R
.
Note that if
u
is a unit in
R
, it cannot be a zero divisor (if
ub
= 0
R
, then multiplying both sides
of this equation by
u

1
yields
b
= 0
R
). In particular, it follows that any field is an integral domain.
36