Is any UFD also a PID?

abstract-algebra ring-theory commutative-algebra unique-factorization-domains principal-ideal-domains

Is there any counterexample that will disprove that every unique factorization domain (UFD) is also a principal ideal domain (PID)? I mean, any PID is a UFD, does the converse hold?

Thanks in advance!


A PID is always of dim $1$. So you can find a lot of UFD's that are not PID; for example:
Every polynomial ring in more than one variable with coeffcients in a field.
Every regular local ring of dim greater than $1$ (i.e the maximal ideal can not generated with one element).

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