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I am reading Dummit and Foote, Section 10 on tensor products of modules.
I am at present trying to understand the use of Theorem 6 (D&F, page 354 - see attachment) in Theorem 8 (D&F page 362, see attachment).
The proof of Theorem 8 in D&F Chapter 10 (see attachment) reads as follows:
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Proof: Suppose $$ \phi : \ N \to L $$ is an R-module homomorphism to the S-module L.
By the universal property of free modules (Theorem 6 in Section 3) there is a Z-module homomorphism from the free Z-module F on the set S x N to L that sends each generator $$(s,n) $$ to $$ s \phi (n) $$. ... ...
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I am trying to see how the use of Theorem 6 applies and operates in the proof of Theorem 8 - so I am essentially trying to "tie up" or "tie together" the symbols and structures of Therem 6 with those of Theorem 8.
Now we know that in Theorem 8 " ... there is a Z-module homomorphism from the free Z-module F on the set S x N to L ..."
Thus for the map $$ A \to F(A) $$ in Theorem 6
we have the map
$$ S \times N \to F(S \times N) $$ in Theorem 8, but this map is not showing in the diagram for Theorem 8 on page 362 (see attachment). ?
To link up to $$ S \oplus_R N = F(S \times N)/H $$ we would need to extend our map above from $$ F(S \times N) $$ to $$ F(S \times N)/H $$ and then apply $$ \Phi $$ to map to L, thus:
$$ S \times N \to F(S \times N) \to F(S \times N)/H = S \oplus_R \to L $$
But how does this fit with the figure for Theorem 8 on page 362 which shows the maps
$$ N \to S \oplus_R N \to L $$ and $$ N \to L $$
So basically I am asking for guidance and help on how the use of Theorem 6 "works" in Theorem 8. In other words, how exactly does it apply? What elements does it apply to and how are its preconditions satisfied?
Hoping someone can help.
Peter
I am at present trying to understand the use of Theorem 6 (D&F, page 354 - see attachment) in Theorem 8 (D&F page 362, see attachment).
The proof of Theorem 8 in D&F Chapter 10 (see attachment) reads as follows:
-------------------------------------------------------------------------------
Proof: Suppose $$ \phi : \ N \to L $$ is an R-module homomorphism to the S-module L.
By the universal property of free modules (Theorem 6 in Section 3) there is a Z-module homomorphism from the free Z-module F on the set S x N to L that sends each generator $$(s,n) $$ to $$ s \phi (n) $$. ... ...
----------------------------------------------------------------------------
I am trying to see how the use of Theorem 6 applies and operates in the proof of Theorem 8 - so I am essentially trying to "tie up" or "tie together" the symbols and structures of Therem 6 with those of Theorem 8.
Now we know that in Theorem 8 " ... there is a Z-module homomorphism from the free Z-module F on the set S x N to L ..."
Thus for the map $$ A \to F(A) $$ in Theorem 6
we have the map
$$ S \times N \to F(S \times N) $$ in Theorem 8, but this map is not showing in the diagram for Theorem 8 on page 362 (see attachment). ?
To link up to $$ S \oplus_R N = F(S \times N)/H $$ we would need to extend our map above from $$ F(S \times N) $$ to $$ F(S \times N)/H $$ and then apply $$ \Phi $$ to map to L, thus:
$$ S \times N \to F(S \times N) \to F(S \times N)/H = S \oplus_R \to L $$
But how does this fit with the figure for Theorem 8 on page 362 which shows the maps
$$ N \to S \oplus_R N \to L $$ and $$ N \to L $$
So basically I am asking for guidance and help on how the use of Theorem 6 "works" in Theorem 8. In other words, how exactly does it apply? What elements does it apply to and how are its preconditions satisfied?
Hoping someone can help.
Peter