[FOM] UFD example

Joachim Reineke reineke at math.uni-hannover.de
Tue Nov 20 12:03:36 EST 2007

Let D = Q[x,y]P be the localisation of the polynomialring over a countable 
field Q with P =(x,y). Then D is a local countable UFD not elementary 
equivalent to a polynomialring R[x], since
R[x] is never a local ring.

----- Original Message ----- 
From: <joeshipman at aol.com>
To: <FOM at cs.nyu.edu>; <RLK at knighten.org>
Sent: Tuesday, November 20, 2007 4:11 PM
Subject: [FOM] UFD example

> 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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