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I am reading Dummit and Foote, CH 10 Section 10.5, Exact Sequences - Projective, Injective and Flat Modules.
As they introduce split sequences, D&F write the following:View attachment 2462
I am concerned at the following statement:
"In this case the module $$ B $$ contains a sub-module $$C'$$ isomorphic to $$C$$ (namely $$ C' = 0 \oplus C$$) as well as the submodule A, and this submodule complement to A "splits" B into a direct sum ... ... "
Maybe I am being pedantic or I am just confused but it does not seem to me that B contains A as a sub-module or that C' is complement to A (regarding A as an abelian group).
It seems to me that if $$ B = A \oplus B $$ then certainly B contains both $$ A' = A \oplus 0 $$ (and NOT A) and $$ C' = 0 \oplus C$$.
To check that the submodule C' is complement to A' we note that
(1) $$ B = A' + C' $$
since
$$ A' + C' = \{ (a,0) + (0,c) = (a,c) | a \in A, c \in C \} $$
and we also note that
(2) $$ A' \cap C' = (0,0) $$
Given (1) and (2), the submodule $$C'$$ is complement to the submodule $$A'$$
Can someone please indicate whether my analysis is correct, and also comment of the D&F text? Should D&F be talking about $$A'$$ being a submodule and specifying $$C'$$ as complement to $$A'$$ (not $$A$$)?
Peter
As they introduce split sequences, D&F write the following:View attachment 2462
I am concerned at the following statement:
"In this case the module $$ B $$ contains a sub-module $$C'$$ isomorphic to $$C$$ (namely $$ C' = 0 \oplus C$$) as well as the submodule A, and this submodule complement to A "splits" B into a direct sum ... ... "
Maybe I am being pedantic or I am just confused but it does not seem to me that B contains A as a sub-module or that C' is complement to A (regarding A as an abelian group).
It seems to me that if $$ B = A \oplus B $$ then certainly B contains both $$ A' = A \oplus 0 $$ (and NOT A) and $$ C' = 0 \oplus C$$.
To check that the submodule C' is complement to A' we note that
(1) $$ B = A' + C' $$
since
$$ A' + C' = \{ (a,0) + (0,c) = (a,c) | a \in A, c \in C \} $$
and we also note that
(2) $$ A' \cap C' = (0,0) $$
Given (1) and (2), the submodule $$C'$$ is complement to the submodule $$A'$$
Can someone please indicate whether my analysis is correct, and also comment of the D&F text? Should D&F be talking about $$A'$$ being a submodule and specifying $$C'$$ as complement to $$A'$$ (not $$A$$)?
Peter
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