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Math 5125
Monday, October 17
Seventh Homework Solutions
1. Note that
n
Z
⊆
m
Z
because
m
±
±
n
. By the ideal correspondence theorem we now obtain
Z
/
n
Z
m
Z
/
n
Z
∼
=
Z
/
m
Z
.
However the ideals
m
Z
/
n
Z
and
m
(
Z
/
n
Z
)
of
Z
/
n
Z
are equal, because they both con
sist of the cosets
mr
+
n
Z
for some
r
∈
Z
, and the result follows.
2. Section 7.3, Exercise 35 on page 250. Let
I
,
J
,
K
be ideals of
R
.
(a) Prove that
I
(
J
+
K
) =
IJ
+
IK
and
(
I
+
J
)
K
=
IK
+
JK
.
(b) Prove that if
J
⊆
I
, then
I
∩
(
J
+
K
) =
J
+(
I
∩
K
)
.
(a) Obviously
IJ
,
IK
⊆
I
(
J
+
K
)
and since
I
(
J
+
K
)
is closed under addition, we see
that
IJ
+
IK
⊆
I
(
J
+
K
)
. On the other hand the general element of
I
(
J
+
K
)
is of
the form
∑
r
i
r
(
j
r
+
k
r
)
, where
i
r
∈
I
,
j
r
∈
J
and
k
r
∈
K
, and this can be written in
the form
∑
r
i
r
j
r
+
∑
r
i
r
k
r
, which shows that
I
(
J
+
K
)
⊆
IJ
+
IK
and it follows that
I
(
J
+
K
) =
IJ
+
IK
. The proof that
(
I
+
J
)
K
=
IK
+
JK
is exactly similar.
(b) Obviously
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This note was uploaded on 01/02/2012 for the course MATH 5125 taught by Professor Palinnell during the Fall '07 term at Virginia Tech.
 Fall '07
 PALinnell
 Algebra, Sets

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