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290
6. RINGS
Example 6.4.2.
(a)
The ring of integers
Z
is an integral domain.
(b)
Any ﬁeld is an integral domain.
(c)
If
R
is an integral domain, then
RŒxŁ
is an integral domain. In
particular,
KŒxŁ
is an integral domain for any ﬁeld
K
.
(d)
If
R
is an integral domain and
R
0
is a subring containing the
identity, then
R
0
is an integral domain.
(e)
The ring of formal power series
RŒŒxŁŁ
with coefﬁcients in an
integral domain is an integral domain.
There are two common constructions of ﬁelds from integral domains.
One construction is treated in Corollary
6.3.14
and Exercise
6.3.7
. Namely,
if
R
is a commutative ring with
1
and
M
is a maximal ideal, then the
quotient ring
R=M
is a ﬁeld.
Another construction is that of the
ﬁeld of fractions
1
of an integral do
main. This construction is known to you from the formation of the rational
numbers as fractions of integers. Given an integral domain
R
, we wish
to construct a ﬁeld from symbols of the form
a=b
, where
a;b
2
R
and
b
¤
0
. You know that in the rational numbers,
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 Fall '08
 EVERAGE
 Algebra, Integers

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