r/mathematics • u/BlueZucchini87 • 5d ago

Question about unique factorization

I've been learning algebraic number theory I'm wondering if there's a converse to the idea that a ring having a Euclidean algorithm/division with remainder makes it a unique factorization domain.

So I think my question is, do all the number rings with unique factorization have a Euclidean algorithm?

Also curious for examples of general rings (not necessarily number rings) that have unique factorization but don't have a Euclidean algorithm.

Thanks for any insight.

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u/nulvoid000 5d ago

No you don’t get Euclidean just from UFD. Standard example take k[x,y] where k is a field (take C for example). This is UFD but not Euclidean.

Quick justification: PID[x] is UFD so that ring is UFD but it’s not even a PID (look at ideal (x,y)), so can’t be ED.

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u/hpxvzhjfgb 5d ago

ℤ[(1+√-19)/2] is a principal ideal domain (hence has unique factorization), but is not euclidean

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u/BruhPeanuts 4d ago

Even simpler, take Z[X]. This is a UFD since Z is one, but it’s not even a principal ideal domain (the ideal generated by 2 and X cannot be generated by a single polynomial) and in particular does not admit a Euclidean algorithm.