Math 5125
Wednesday, November 9
November 9, Ungraded Homework
Exercise 10.3.12 on page 356
Let
R
be a commutative ring with a 1 and let
A
,
B
and
M
be
R
modules. Prove the following isomorphisms of
R
modules.
(a) Hom
R
(
A
⊕
B
,
M
)
∼
=
Hom
R
(
A
,
M
)
⊕
Hom
R
(
B
,
M
)
.
(b) Hom
R
(
M
,
A
⊕
B
)
∼
=
Hom
R
(
M
,
A
)
⊕
Hom
R
(
M
,
B
)
.
The proofs of (a) and (b) are similar; we will only prove (a).
If
γ
∈
Hom
R
(
A
⊕
B
,
M
)
, define
α
(
γ
)
∈
Hom
R
(
A
,
M
)
by
α
(
γ
)(
a
) =
γ
(
a
,
0
)
and
β
(
γ
)
∈
Hom
R
(
B
,
M
)
by
β
(
γ
)(
b
) =
γ
(
0
,
b
)
.
If
δ
∈
Hom
R
(
A
⊕
B
,
M
)
, then
α
(
γ
+
δ
)(
a
) = (
γ
+
δ
)(
a
,
0
) =
γ
(
a
,
0
)+
δ
(
a
,
0
) =
α
(
γ
)(
a
)+
α
(
δ
)(
a
) = (
α
(
γ
)+
α
(
δ
))(
a
)
for all
a
∈
A
and we
see that
α
(
γ
+
δ
) =
α
(
γ
)+
α
(
δ
)
. Also if
r
∈
R
, then
(
α
(
r
γ
))(
a
) = (
r
γ
)(
a
,
0
) =
γ
(
ra
,
0
) =
(
α
(
γ
))(
ra
) = (
r
(
α
(
γ
)))(
a
)
for all
a
∈
A
, hence
α
(
r
γ
) =
r
(
α
(
γ
))
and we deduce that
α
is
an
R
module homomorphism. Similarly
β
is an
R
module homomorphism. Finally we de
fine an
R
module homomorphism
θ
: Hom
R
(
A
⊕
B
,
M
)
→
Hom
R
(
A
,
M
)
⊕
Hom
R
(
B
,
M
)
by
θ
(
γ
) = (
α
(
γ
)
,
β
(
γ
))
.
Now we want to define the inverse map to
θ
. If
f
∈
Hom
R
(
A
,
M
)
, define
f
∈
Hom
R
(
A
⊕
B
,
M
)
by
f
(
a
,
b
) =
f
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 Fall '07
 PALinnell
 Math, Algebra, φ, Abelian group, θ, φ θ, HomR

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