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2000 Nov

2000 November 10

  1. Let \(I\) be an ideal of a ring \(R\).\ a. Describe (without proofs) a one-to-one correspondence between the set of ideals of \(R\) containing \(I\) and the set of ideals of \(R/I\).\ b. Prove that the correspondence matches prime ideals to prime ideals.\ c. Prove that an ideal \(I\) of a commutative ring \(R\) is maximal if and only if \(R/I\) is a field.

  2. Prove that every finite integral domain is a field.

  3. If \(F\) is a field, prove that the polynomial ring \(F[x]\) is a principal ideal domain. Is the same true for \(\mathbb Z[x]\)? Why?

  4. If \(D\) is a principal ideal domain, but not a field, then prove that \(D\) satisfies the ascending chain condition for ideals.

  5. a. If \(I\) and \(J\) are ideals in a commutative ring \(R\) such that \(I + J = R\), then prove that \(IJ = I \cap J\).\ b. With \(R\) commutative, prove that the set of non-units forms an ideal of \(R\) if and only if \(R\) contains a unique maximal ideal, and give an example of such an \(R\).