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- Jun 22, 2012

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I am reading Dummit and Foote, Section 10.4: Tensor Products of Modules. I am studying Corollary 9 and attempting to fully understand the Corollary and it proof. (For details see the attachement page 362 in which Theorem 8 is stated and proved. This is followed by the statement and proof of Corollary 9.

The proof of Corollary 9 reads as follows:

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Suppose now that \(\displaystyle \phi \) is an R-module homomorphism injecting the quotient \(\displaystyle N/ker \ \phi \) of N into an S-module L.

Then, by Theorem 8, ker i is mapped to 0 by \(\displaystyle \phi \), that is \(\displaystyle ker \ i \subseteq ker \ \phi \).

... ... etc etc

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I do not fully understand how D&F reached the conclusion that \(\displaystyle ker \ i \subseteq ker \ \phi \)

A simple diagram showing the maps of Corollary 9 is attached.

Could someone also clarify the following issue for me:

In corollary 9 D&F refer to "the quotient \(\displaystyle N/ker \ \phi \) of \(\displaystyle N\) ... ... does this actually mean the coset of the quotient module \(\displaystyle N/ker \ \phi \) or are they referring to the quotient module? (Hope I am making myself clear - I am a bit confused by the term ... )

Peter

The proof of Corollary 9 reads as follows:

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**Proof:**The quotient\(\displaystyle N/ ker \ i \) is mapped injectively (by i) into the S-module \(\displaystyle S \oplus_R N\).Suppose now that \(\displaystyle \phi \) is an R-module homomorphism injecting the quotient \(\displaystyle N/ker \ \phi \) of N into an S-module L.

Then, by Theorem 8, ker i is mapped to 0 by \(\displaystyle \phi \), that is \(\displaystyle ker \ i \subseteq ker \ \phi \).

... ... etc etc

----------------------------------------------------------------------------

I do not fully understand how D&F reached the conclusion that \(\displaystyle ker \ i \subseteq ker \ \phi \)

**Can someone show me (formally and rigorously) why, as D&F assert, by Theorem 8, it follows that \(\displaystyle ker \ i \subseteq ker \ \phi \)?**(Edit : I suppose this reduces to the question of why, exactly, ker i is mapped to zero by \(\displaystyle \phi \).)A simple diagram showing the maps of Corollary 9 is attached.

Could someone also clarify the following issue for me:

In corollary 9 D&F refer to "the quotient \(\displaystyle N/ker \ \phi \) of \(\displaystyle N\) ... ... does this actually mean the coset of the quotient module \(\displaystyle N/ker \ \phi \) or are they referring to the quotient module? (Hope I am making myself clear - I am a bit confused by the term ... )

Peter

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