Sunday, May 12, 2013
Characterization of PIDs (8.3.11)
Dummit and Foote Abstract Algebra, section 8.3, exercise 11:
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ProveR a Principle Ideal Domain ⇔R a UFD ∧R a Bezout DomainProof: The ⇒ direction is evident, so it suffices to demonstrate the converse. Observe a typical ideal I⊆R, and choose arbitrary x∈I with x=uπα11...παnn being its unique decomposition up to units. For any e∈Z+∪{0} and i with 1≤i≤n let Ai,e={r∈I | πei∣r}. For each i let βi be the largest integer such that Ai,βi=I, which is guaranteed to exist for arbitrary i as the exponent αi of πi is finite so that as a bound x∉Ai,αi+1. For each i, choose ai such that ai∈I∖Ai,βi+1, so that ai=πβiini with πi∤ni. Letting A=⋃1≤i≤nai and taking successive greatest common divisors of the ai and then taking a greatest common divisor of this and x, we have a final common divisor d=πβ11...πβnn (since these are the only irreducibles dividing x, and for each i there is an element ai with its minimal πi exponent βi, and all the exponents of the aj and x of πi is at least βi as aj,x∈I=Ai,βi), writable as an R-linear combination of elements from A∪{x}⊆I by 7.2.7. We claim (d)=I: Evidently (d)⊆I, and for the reverse containment, for any r∈I we have πβii∣r for applicable i since r∈I=Ai,βi, implying I⊆(d) and now I=(d) is a principle ideal. ◻
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