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Sunday, May 12, 2013

Characterization of PIDs (8.3.11)

Dummit and Foote Abstract Algebra, section 8.3, exercise 11:

MathJax TeX Test Page 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 IR, and choose arbitrary xI with x=uπα11...παnn being its unique decomposition up to units. For any eZ+{0} and i with 1in let Ai,e={rI | πeir}. 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 xAi,αi+1. For each i, choose ai such that aiIAi,βi+1, so that ai=πβiini with πini. Letting A=1inai 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,xI=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 rI we have πβiir for applicable i since rI=Ai,βi, implying I(d) and now I=(d) is a principle ideal. 

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