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

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SUMMARY

The discussion centers on the proof of Theorem 7 from Dummit and Foote's "Abstract Algebra," specifically in Section 9.3, which states that a ring R is a Unique Factorization Domain (UFD) if and only if the polynomial ring R[x] is also a UFD. The key argument presented is that if R[x] is a UFD, then the constant polynomials, which correspond to the elements of R, must also have unique factorization, thereby establishing that R itself is a UFD. This logical connection is crucial for understanding the properties of polynomial rings in relation to their base rings.

PREREQUISITES
  • Understanding of Unique Factorization Domains (UFDs)
  • Familiarity with polynomial rings, specifically R[x]
  • Knowledge of the concepts presented in Dummit and Foote's "Abstract Algebra"
  • Basic proof techniques in abstract algebra
NEXT STEPS
  • Study the properties of Unique Factorization Domains in detail
  • Review the proof techniques used in Dummit and Foote's "Abstract Algebra"
  • Explore examples of polynomial rings that are UFDs
  • Learn about the implications of UFDs in algebraic geometry
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Students and educators in abstract algebra, particularly those studying polynomial rings and their properties, as well as anyone preparing for advanced mathematics examinations that cover UFDs.

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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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