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2003 Nov

2003 November

  1. Let \(R\) be a commutative ring, and let \(N\) be the set of all nilpotent elements of \(R\). Prove that\ a. \(N\) is an ideal of \(R\), and\ b. \(R/N\) has no nonzero nilpotent elements.

  2. a. Show that \(\mathbb Z[\sqrt{-1}]\) is a unique factorization domain (UFD).\ b. Show that \(\mathbb Z[\sqrt{-3}]\) is not a UFD.

  3. Let \(R\) be a commutative ring with unit. Show that if \(R\) contains an idempotent element \(e\), then there exist ideals \(S\), \(T\) of \(R\) such that \(R = S \oplus T\).

  4. Prove that if \(R\) is a commutative ring and \(I, J\) are ideals of \(R\), then there is an \(R\)-module isomorphism \(R/I \otimes R/J \cong R/(I+J)\).

  5. Let \(M\) be a left \(R\)-module and \(x\in M\). Let \(\mathrm{ann}(x) = \{r \in R : rx = 0\}\).\ a. Show that \(\mathrm{ann}(x)\) is a left ideal of \(R\).\ b. Prove that there is an \(R\)-module isomorphism \(Rx \cong R/\mathrm{ann}(x)\).