Question: 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

4. Let M be an R-module and ze R. Prove the following:

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 IR R/I 0 and the long exact sequence theorem for Tor to prove that Torf (R/I, R/J) TJ

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