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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 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=IJ.
    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.