Page 138 - Advanced Linear Algebra
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122 Advanced Linear Algebra
4~ 5 l ker ² ³
To see this, we have
5 q ker ² ³ ~ ker ² ³ ~ ¸ ¹
and if 4 , then there is a 5 for which ~ and so
² c ³ ~ c ~
Thus,
~ b ² c ³ 5 b ker ² ³
5
which shows that ker²³ is a complement of .
Theorem 4.14 Let 4 and 4 be -modules and let 5 4 .
9
)
1 If 4~ 5 l / , then any 9 -epimorphism ¢ 5 ¦ 4 has a unique
to an epimorphism with
extension ¢4 ¦ 4
ker²³ ~ ker²³ l /
) be an -isomorphism. Then the correspondence
2 Let ¢5 4 9
,
/ª where ker ² ³ ~ /
is a bijection from complements of 5 onto the extensions of . Thus, an
isomorphism ¢5 4 has an extension to 4 if and only if 5 is
complemented.
Definition Let 5 4 . When the identity map ¢ 5 5 has an extension to
¢4 ¦ 5, the submodule
5 is called a retract of 4 and is called the
retraction map.
5
Corollary 4.15 A submodule 5 4 is a retract of 4 if and only if has a
complement in 4 .
Direct Summands and One-Sided Invertibility
Direct summands are also related to one-sided invertibility of -maps.
9
Definition Let ¢( ¦ ) be a module homomorphism.
)
1 A left inverse of is a module homomorphism 3 ¢) ¦ ( for which
3 k ~ .
2 A right inverse of is a module homomorphism 9 ¢) ¦ ( for which
)
k ~ 9 .
Left and right inverses are called one-sided inverses . An ordinary inverse is
called a two-sided inverse .
Unlike a two-sided inverse, one-sided inverses need not be unique.
