Math 5125
Wednesday, November 30
November 30, Ungraded Homework
Exercise 10.5.4 on page 403
Let
R
be a ring with a 1 and let
Q
1
and
Q
2
be
R
modules.
Prove that
Q
1
⊕
Q
2
is an injective
R
module if and only if
Q
1
and
Q
2
are injective.
Suppose
Q
1
⊕
Q
2
is an injective
R
module; we will prove that
Q
1
must also be injective.
Suppose we are given a diagram of
R
modules and
R
maps with exact row
0
→
A
α
→
B
y
θ
Q
1
We want to ﬁnd an
R
map
φ
:
B
→
Q
1
making the diagram commute, that is
φα
=
θ
. Deﬁne
θ
1
:
A
→
Q
1
⊕
Q
2
by
θ
1
(
a
) = (
θ
a
,
0
)
. Then clearly
θ
1
is an
R
map, so by injectivity of
Q
1
⊕
Q
2
, there exists an
R
map
φ
1
:
B
→
Q
1
⊕
Q
2
such that
αφ
1
=
θ
1
. Let
π
1
:
Q
1
⊕
Q
2
→
Q
1
denote the natural projection, so
π
1
(
q
) = (
q
,
0
)
, and deﬁne
φ
=
π
1
φ
1
:
B
→
Q
1
. Then
φα
(
a
) =
π
1
φ
1
α
(
a
) =
π
1
θ
1
(
a
) =
π
1
(
θ
a
,
0
) =
θ
(
a
)
,
so
φα
=
θ
and we have shown that
Q
1
is injective. Similarly
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 Fall '07
 PALinnell
 Math, Algebra, Natural number, Universal quantification, Existential quantification, Finite set

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