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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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, Vector Space

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