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6.4. INTEGRAL DOMAINS
289
6.3.9.
For any ring
R
, and any natural number
n
, we can deﬁne the matrix
ring Mat
n
.R/
consisting of
n
by
n
matrices with entries in
R
. If
J
is an
ideal of
R
, show that Mat
n
.J/
is an ideal in Mat
n
.R/
and furthermore
Mat
n
.R/=
Mat
n
.J/
Š
Mat
n
.R=J/
.
Hint:
Find a natural homomorphism
from Mat
n
.R/
onto Mat
n
.R=J/
with kernel Mat
n
.J/
.
6.3.10.
Let
R
be a commutative ring. Show that
RŒxŁ=xRŒxŁ
Š
R
.
6.3.11.
This exercise gives a version of the
Chinese remainder theorem
.
(a)
Let
R
be a ring,
P
and
Q
ideals in
R
, and suppose that
P
\
Q
D
f
0
g
, and
P
C
Q
D
R
. Show that the map
x
7!
.x
C
P;x
C
Q/
is an isomorphism of
R
onto
R=P
˚
R=Q
.
Hint:
Injectivity is
clear. For surjectivity, show that for each
a;b
2
R
, there exist
x
2
R
,
p
2
P
, and
q
2
Q
such that
x
C
p
D
a
, and
x
C
q
D
b
.
(b)
More generally, if
P
C
Q
D
R
, show that
R=.P
\
Q/
Š
R=P
˚
R=Q
.
6.3.12.
(a)
Show that integers
m
and
n
are relatively prime if, and only if,
m
Z
C
n
Z
D
Z
if, and only if,
m
Z
\
n
Z
D
mn
Z
. Conclude that
if
m
and
n
are relatively prime, then
Z
mn
Š
Z
m
˚
Z
n
as rings.
(b)
State and prove a generalization of this result for the ring of poly
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
 Fall '08
 EVERAGE
 Algebra, Matrices

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