Integral domains and ideals
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Let R be an integral domain and suppose that every prime
ideal in R is principal. This exercise proves that every ideal of R is principal.
(a) Assume that the set of ideals of R that are not principal is nonempty
and prove that this set has a maximal element. [Use Zorn's Lemma.]
(b) 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: Let I_a = (I,a) be the
ideal generated by I and a, let I_b = (I,b) be the ideal generated by
I and b, and define J = {r in R/ rI_ai is contained in I}. Prove that I_a = (alpha)
J = (beta) are principal ideals in R with I not contained in I_b not contained in J and
I_aJ=(alpha beta) contained in I.
(c) If x is in I show that x = s alpha for some s in J: Deduce that I = I_aJ is
principal, a contradiction, and conclude that R is a PID(principal ideal domain)
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Solution Summary
This is a set of proofs regarding integral domains.
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