Page 40 - Advanced Linear Algebra

24 Advanced Linear Algebra

p ~ mod ~
and so and are both zero divisors. As we will see later, if is a prime, then

{ is a field which is an integral domain, of course .( )
Example 0.15 The ring -´%µ is an integral domain, since ²%³ ²%³ ~ implies
that ²%³ ~ or ²%³ ~ .

9
If is a ring and % ~ & where Á %Á &  9 , then we cannot in general cancel

the 's and conclude that % ~ & . For instance, in { , we have h ~ h , but

canceling the 's gives ~ . However, it is precisely the integral domains in
which we can cancel. The simple proof is left to the reader.
Theorem 0.24 Let be a commutative ring with identity. Then is an integral
9
9
domain if and only if the cancellation law
% ~ &Á £ ¬ % ~ &
holds.
The Field of Quotients of an Integral Domain

Any integral domain can be embedded in a field. The quotient field or field
(
9
of quotients) of is a field that is constructed from just as the field of
9
9
rational numbers is constructed from the ring of integers. In particular, we set
b
9 ~ ¸² Á ³ “ Á  9Á £ ¹
Z
Z
Z
Z
where ² Á ³ ~ ² Á ³ if and only if ~ . Addition and multiplication of
fractions is defined by
² Á ³ b ² Á ³ ~ ² b Á ³
and
² Á ³ h ² Á ³ ~ ² Á ³
It is customary to write ² Á ³ in the form ° . Note that if has zero divisors,
9

then these definitions do not make sense, because may be even if and
are not. This is why we require that be an integral domain.
9
Principal Ideal Domains
Definition Let be a ring with identity and let  9 . The principal ideal
9
generated by is the ideal

º » ~ ¸ “  9¹

An integral domain 9 in which every ideal is a principal ideal is called a
principal ideal domain.