Originally Posted by

**robeuler** Hi everyone

I am having trouble with this problem. It is number 6 in section 8.2 of Dummit and Foote if you have it.

R is an integral domain and suppose that every prime ideal is principal.

a) (I have done this) Assume that the set of ideals of R that are not principal is nonempty and prove that this set has a maximal element under inclusion.

b) (Where I am stuck) Let I be an ideal which is maximal with respect to being nonprincipal, and let a, b in R with

ab in I, but a not in I and b not in I (so I is not a prime ideal)

Let Ia be the ideal generated by (I,a) and Ib be the ideal generated by (I,b). Define J={r in R| rIa is a contained in I}

Prove that Ia=(v) and J=(w) (generated by v and w, respectively) are principal ideals in R with I properly contained in Ib contained in J and IaJ=(vw) contained in I.

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First with Ia, I cannot figure out how to express Ia as being generated by v. I also tried to suppose that it is not principal, but haven't gotten anywhere. Any ideas?