4. Let M be an R-module and ze R. Prove the following: (a) There is a...

 

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4. Let M be an R-module and ze R. Prove the following: (a) There is a short exact sequence 0 → R/(0 :Rx) $ R→ R/TR → 0, where j is induced by multiplication by a, and the other nontrivial arrow is the canonical surjection. (b) Prove that Tor (R/TR, M) = (M) I (c) Use the short exact sequence in part (a) and the long exact sequence theorem for Tor to prove that Tor (R/xR, M) = Torn-1(R/(0 :R x), M) for all n ≥ 2. 5. Let I, J be ideals of R. Use the short exact sequence 0 I→R → R/I → 0 and the long exact sequence theorem for Tor to prove that Torf (R/I, R/J) TJ 4. Let M be an R-module and ze R. Prove the following: (a) There is a short exact sequence 0 → R/(0 :Rx) $ R→ R/TR → 0, where j is induced by multiplication by a, and the other nontrivial arrow is the canonical surjection. (b) Prove that Tor (R/TR, M) = (M) I (c) Use the short exact sequence in part (a) and the long exact sequence theorem for Tor to prove that Tor (R/xR, M) = Torn-1(R/(0 :R x), M) for all n ≥ 2. 5. Let I, J be ideals of R. Use the short exact sequence 0 I→R → R/I → 0 and the long exact sequence theorem for Tor to prove that Torf (R/I, R/J) TJ