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