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

[Math Amateur]

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]

 

[ModusPonens]

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?