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Homework X: August 4, 2009
3
.
51
(
i
)
.
If
A
and
R
are domains and
ϕ
:
A
→
R
is a ring isomorphism, prove
that Φ: [
a, b
]
7→
[
fa, fb
] is a ring isomorphism Frac(
A
)
→
Frac(
R
).
Note ﬁrst that since
f
is injective,
b
6
= 0 implies
fb
6
= 0; hence, [
fa, fb
] makes
sense. Next, Φ is welldeﬁned: if [
a, b
] = [
c, d
], then [
fa, fb
] = [
fc, fd
]. This
is true: since
ad
=
bc
, we have
fafd
=
f
(
ad
) =
f
(
bc
) =
fbfc
. We check that
Φ is a ring map. Φ[1
,
1] = [
f
1
, f
1] = [1
,
1]; Φ([
a, b
] + [
c, d
]) = Φ[
ad
+
bc, bd
] =
[
f
(
ad
+
bc
)
, f
(
bd
)] = [
fafd
+
fbfc, fbfd
] = [
fa, fb
] + [
fc, fd
] = Φ[
a, b
] + Φ[
c, d
].
Similarly, Φ preserves multiplication. To see that Φ is an isomorphism, we
exhibit its inverse Φ

1
. Deﬁne Ψ: Frac(
R
)
→
Frac(
A
) by [
r, s
]
7→
[
f

1
r, f

1
s
].
(
ii
)
.
Show that a ﬁeld
k
containing an isomorphic copy of
Z
as a subring must
contain an isomorphic copy of
Q
.
By hypothesis, there is an isomorphism
f
:
Z
→
R
, where
R
is a subring of
k
. It follows from (i) that there is an isomorphism
f
: Frac(
Z
)
→
R
0
, where
R
0
is a subﬁeld of
k
containing im
f
. But Frac(
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This note was uploaded on 11/15/2010 for the course MATH 417 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.
 Spring '08
 Staff
 Algebra

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