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Math Help - Polynomial Rings and UFDs - Dummit and Foote pages 303-304

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    Super Member Bernhard's Avatar
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    Polynomial Rings and UFDs - Dummit and Foote pages 303-304

    I am reading Dummit and Foote Section 9.3 Polynomial Rings That are Unique Factorization Domains (see attachment Section 9.3 pages 303 -304)

    I am working through (beginning, anyway) the proof of Theorem 7 which states the following:

    "R is a Unique Factorization Domain if and only if R[x] is a Unique Factorization Domain"

    The proof begins with the statement:

    Proof: We have indicated above the R[x] a Unique Factorization Domain forces R to be a Unique Factorization Domain"

    Question! Can anyone explain to me how and where D&F indicate or explain this first statement of the proof. Secondly, what is the explanation

    I have provided Section 9.3 up to Theorem 7 as an attachment.

    Would appreciate some help.

    Peter
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    Last edited by Bernhard; May 14th 2013 at 08:43 PM.
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    Re: Polynomial Rings and UFDs - Dummit and Foote pages 303-304

    It's easier to see the contrapositive: If R is not a UFD, how can R[x] possibly one? If you think of elements of R as constant polynomials, this should be trivial.
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