1
Homework IX: July 31, 2009
3
.
9
.
Find all the units in the commutative ring
R
=
F
(
R
) in Example 3.11(i).
We claim that
f
is a unit in
R
if and only if
f
(
r
) = 0 for all
r
∈
R
. If
f
is
a unit, then there is
g
∈
R
with
fg
= 1; that is, (
fg
)(
r
) =
f
(
r
)
g
(
r
) = 1 for all
r
∈
R
. Hence,
f
(
r
) = 0. Conversely, suppose that
f
(
r
) = 0 for all
r
∈
R
. Define
g
by
g
(
r
) =
(
1
/f
(
r
)
)
for each
r
. Then (
fg
)(
r
) =
f
(
r
)
g
(
r
) = 1 for all
r
; hence,
fg
= 1, and so
f
is a unit.
3
.
19
.
Define
F
4
to be the set of all 2
×
2 matrices
a
b
b a
+
b
, where
a, b
∈
I
2
.
(
i
)
.
Prove that
F
4
is a commutative ring whose operations are matrix addition
and matrix multiplication.
Clearly, the identity matrix
I
lies in
F
4
.
F
4
is closed under matrix addition:
a
b
b a
+
b
+
c
d
d c
+
d
=
a
+
c
b
+
d
b
+
d a
+
c
+
b
+
d
;
F
4
is closed under matrix multiplication.
a
b
b a
+
b
c
d
d c
+
d
=
ac
+
bd
ad
+
bc
+
bd
ac
+
bd
+
bd bd
+(
a
+
b
)(
c
+
d
)
.
This formula also shows that
F
4
is commutative. All the other properties we
need: associativity, distributivity, etc. hold because
F
4
is a subring of ring of all
2
×
2 matrices with entries in
I
4
. I will allow you to refer to the usual properties
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 Spring '08
 Staff
 Algebra, Ring, Commutative ring, ak xk

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