270
6. RINGS
6.1.11.
Show that the set
R
.x/
of rational functions
p.x/=q.x/
, where
p.x/; q.x/
2
R
OExŁ
and
q.x/
¤
0
, is a field. (Note the use of parentheses
to distinguish this ring
R
.x/
of rational functions from the ring
R
OExŁ
of
polynomials.)
6.1.12.
Let
R
be a ring and
X
a set.
Show that the set Fun
.X; R/
of
functions on
X
with values in
R
is a ring. Show that
R
is isomorphic to the
subring of constant functions on
X
. Show that Fun
.X; R/
is commutative
if, and only if,
R
is commutative. Suppose that
R
has an identity; show
that Fun
.X; R/
has an identity and describe the units of Fun
.X; R/
.
6.1.13.
Let
S
End
K
.V /
, where
V
is a vector space over a field
K
. Show
that
S
0
D f
T
2
End
K
.V /
W
TS
D
ST
for all
S
2
S
g
is a subring of End
K
.V /
.
6.1.14.
Let
V
be a vector space over a field
K
. Let
G
be a subgroup of
GL.V /
. Show that the subring of End
K
.V /
generated by
G
is the set of
all linear combinations
P
g
n
g
g
of elements of
G
, with coefficients in
Z
.
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 Fall '08
 EVERAGE
 Algebra, Polynomials, Rational Functions, Ring, EndK .V

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