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- Jun 22, 2012

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In Example 7 in Dummit and Foote, Section 10.4. pages 369-370 (see attachment) D&F are seeking to establish an isomorphism:

\(\displaystyle S \otimes_R R \cong S \)

They establish the existence of two S-module homomorphisms:

\(\displaystyle \Phi \ : \ S \otimes_R R \to S \)

defined by \(\displaystyle \Phi (s \otimes r ) = sr \)

and

\(\displaystyle {\Phi}' \ : \ S \to S \otimes_R R \)

defined by \(\displaystyle {\Phi}' (s) = s \otimes 1 \)

D&F then show that \(\displaystyle \Phi {\Phi}' = I \) where I is the identity function on simple tensors ...

How does this establish that \(\displaystyle S \otimes_R R \cong S \) ... presumably this establishes \(\displaystyle \Phi \) as a bijective homomorphism ... but how exactly ...

Peter

\(\displaystyle S \otimes_R R \cong S \)

They establish the existence of two S-module homomorphisms:

\(\displaystyle \Phi \ : \ S \otimes_R R \to S \)

defined by \(\displaystyle \Phi (s \otimes r ) = sr \)

and

\(\displaystyle {\Phi}' \ : \ S \to S \otimes_R R \)

defined by \(\displaystyle {\Phi}' (s) = s \otimes 1 \)

D&F then show that \(\displaystyle \Phi {\Phi}' = I \) where I is the identity function on simple tensors ...

How does this establish that \(\displaystyle S \otimes_R R \cong S \) ... presumably this establishes \(\displaystyle \Phi \) as a bijective homomorphism ... but how exactly ...

Peter

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