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
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 EVERAGE
 Algebra, Matrices, Ring, Ring theory, Commutative ring, nonzero elements, Matn .R/

Click to edit the document details