[FOM] UFD example

joeshipman@aol.com joeshipman at aol.com
Tue Nov 20 10:11:44 EST 2007

Bob Knighten points out that formal power series rings in any number of 
variables over a field are UFDs, so this answers my query because with 
more than one variable they are not PIDs.

But what I really wanted (sorry I didn't say so!) was a *countable* 
example of a UFD that was neither a PID nor a polynomial ring.

If start with Q[[x,y]], which is uncountable, by the Lowenheim-Skolem 
theorem this will have a countable elementary submodel. Such a model 
will have the first-order property "irreducibles are prime", which in 
the presence of the descending chain condition implies unique 
factorization, and the DCC condition is preserved in submodels. It will 
also have a nonprincipal ideal generated by 2 elements since that can 
also be expressed as a first-order property.

Unfortunately, I can't express the property of not being a polynomial 
ring as a first-order property, so my submodel might actually be a 
polynomial ring! Can this construction be repaired, or must I find a 
countable example in a different way?

-- JS
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