MHB Polynomial Rings and UFDs - Dummit and Foote pages 303-304

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The discussion centers on the proof of Theorem 7 from Dummit and Foote, which states that if R[x] is a Unique Factorization Domain (UFD), then R must also be a UFD. The initial inquiry seeks clarification on where Dummit and Foote support this assertion in their text. A participant suggests that since the constant polynomials in R[x] correspond to elements in R and possess unique factorization, it follows that R itself must be a UFD. This reasoning aligns with the theorem's proof, confirming the relationship between polynomial rings and unique factorization. The conversation emphasizes the importance of understanding the foundational properties of UFDs in polynomial contexts.
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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 am so far unable to find anything that indicates to me that this is true.

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

Would appreciate some help.

Peter

[This question is also posted on MHF]
 
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Hello

I'm sorry if this is wrong. I haven't seen polynomial rings in at least 8 years. Anyway, I think that the answer is: If R[x] is a UFD, the polynomials that are constants, in particular, have unique factorization. Since the constants of R[x] are isomorphic to R, R is a UFD.

Does this make sense?
 
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