Page 148 - Advanced Linear Algebra
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132 Advanced Linear Algebra
where £ for all and £ for some . Multiplying by ~ Ä £
produces a nontrivial linear dependency over ,
9
# b Ä b # ~
which implies that ~ for all . Thus is linearly dependent over if and
9
8
only if it is linearly dependent over . But in the vector space 8 ² ³ , all sets of
8
cardinality greater than are linearly dependent over and hence all subsets of
8
²9 ³ of cardinality greater than are linearly dependent over 9.
Free Modules and Epimorphisms
If is a module epimorphism where is free on , then it is easy to
-
8¢4 ¦ -
define a right inverse for , since we can define an -map 9 ¢ - ¦ 4 9 by
specifying its values arbitrarily on and extending by linearity. Thus, we take
8
9 ² ³ to be any member of c ² ³. Then Theorem 4.16 implies that ker ² ³ is a
direct summand of 4 and
4 ker ² ³ ^ -
This discussion applies to the canonical projection ¢4 ¦ 4°: provided that
the quotient 4°: is free.
Theorem 5.6 Let be a commutative ring with identity.
9
¢4 ¦ -
1 If ) is an 9 -epimorphism and - is free, then ker ² ³ is
complemented and
4~ ker ² ³ l 5 ker ² ³ ^ -
where 5 - .
)
2 If is a submodule of 4 and if 4 ° : is free, then is complemented and
:
:
4
4 : ^
:
If 4Á : and 4°: are free, then
4
rk²4³ ~ rk²:³ b rk 6 7
:
and if the ranks are all finite, then
4
rk 6 7 ~ rk² 4 ³ c rk² : ³
:
Noetherian Modules
One of the most desirable properties of a finitely generated -module 4 is that
9
all of its submodules be finitely generated:
