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8.2. HOMOMORPHISMS AND QUOTIENT MODULES
353
8.1.8.
Let
V
be an
n
–dimensional vector space over a ﬁeld
K
. Show that
V
n
(the direct sum of
n
copies of
V
) is a free module over End
K
.V /
.
8.1.9.
Let
V
be a ﬁnite dimensional vector space
V
over a ﬁeld
K
. Let
T
2
End
K
.V /
. Give
V
the corresponding
KŒxŁ
–module structure deﬁned
by
P
i
˛
i
x
i
v
D
P
i
˛
i
T
i
.v/
. Show that
V
is not free as a
KŒxŁ
–module.
8.1.10.
Show that conditions (a) and (c) in Proposition
8.1.28
are equiva
lent.
8.2. Homomorphisms and quotient modules
In this section, we construct quotient modules and develop homomor
phism theorems for modules, which are analogues of the homomorphism
theorems for groups and rings. Recall that if
M
and
N
are
R
–modules,
then Hom
.M;N/
denotes the set of
R
–module homomorphisms from
M
to
N
.
Proposition 8.2.1.
(a)
If
'
2
Hom
R
.M;N/
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 Fall '08
 EVERAGE
 Algebra, Vector Space

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