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100
(ii) Symmetric: If
ϕ
:
A
→
R
is an isomorphism, then part (i)
shows that its inverse is an isomorphism
R
→
A
.
(iii) Transitive: If
A
ϕ
→
B
θ
→
C
be ring homomorphisms, then part (ii) shows that their com
posite is an isomorphism
A
→
C
.
3.45
Let
R
be a commutative ring and let
F
(
R
)
be the commutative ring of all
functions
f
:
R
→
R
.
(i)
Show that
R
is isomorphic to the subring of
F
(
R
)
consisting of
all the constant functions.
Solution.
If
r
∈
R
, let
ε
r
denote the constant function
R
→
R
sending
a
7→
r
for all
a
∈
R
. It is routine to check that
ϕ
:
R
→
F
(
R
)
, given by
r
7→
ε
r
, is an injective homomorphism with the
desired image.
(ii)
If
f
(
x
)
=
a
0
+
a
1
x
+···+
a
n
x
n
∈
R
[
x
]
, let
f
[
:
R
→
R
be de
f
ned
by
f
[
(
r
)
=
a
0
+
a
1
r
+···+
a
n
r
n
[thus,
f
[
is the polynomial
function associated to
f
(
x
)
]. Show that the function
ϕ
:
R
[
x
]→
F
(
R
)
,de
f
ned by
ϕ
:
f
(
x
)
7→
f
[
, is a ring homomorphism.
Solution.
Absent.
(iii)
Show that if
R
=
F
p
, where
p
is a prime, then
x
p
−
x
∈
ker
ϕ
.
Solution.
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 Fall '11
 KeithCornell

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