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For example, integers, gaussian integers, and polynomials all have unique factorizations. What are these rings (or this property) referred to as? Or is unique factorization a ubiquitous property that applies to all rings? If not, what is an example of a ring that doesn't have any meaningful "unique factorization" property?

I'm no expert at abstract algebra, so sorry if this is a silly question, or I'm using the term "ring" incorrectly (perhaps field or group is the more applicable term).

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    $\begingroup$ Paste your title into google and click "I'm feeling lucky" ;-) $\endgroup$
    – dxiv
    May 13, 2022 at 6:36
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    $\begingroup$ @dxiv My bad >.<. Thanks <3 $\endgroup$
    – chausies
    May 13, 2022 at 6:38
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    $\begingroup$ Yes... always worth briefly researching before asking. I am pretty sure you mean this: en.wikipedia.org/wiki/Unique_factorization_domain $\endgroup$
    – FShrike
    May 13, 2022 at 6:38
  • $\begingroup$ Unique factorization (or factorial) ring / domain. Please delete the question since it is a dupe. $\endgroup$
    – Bill Dubuque
    May 13, 2022 at 9:02
  • $\begingroup$ @DietrichBurde yeah, I meant to say "Abstract Algebra", but slipped up and wrote "Algebraic Geometry" without thinking >.<. $\endgroup$
    – chausies
    May 14, 2022 at 7:32

1 Answer 1

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The answer is Unique Factorization Domains. Thanks Google and Wikipedia!

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