[FOM] UFD example?

Joachim Reineke reineke at math.uni-hannover.de
Mon Nov 19 13:53:46 EST 2007

Let K be a field and D := K[[x,y]] the power series ring in two
indeterminate. Then D is not a PID nor isomorphic to a polynomial ring,
since 1 + x has a root (char(K) < > 2) in D, but never in R[x]. Clearly D is 
a UFD.

Mit freundlichen Grüßen

Prof.Dr.Joachim Reineke
Tel.:      0511-762-2890
privat :  05108-1574
----- Original Message ----- 
From: <joeshipman at aol.com>
To: <fom at cs.nyu.edu>
Sent: Sunday, November 18, 2007 3:38 AM
Subject: [FOM] UFD example?

> What is an example of Unique Factorization Domain which is neither a
> Principal Ideal Domain nor (isomorphic to) a polynomial ring? (By a
> polynomial ring I mean a ring R[x] formed by adjoining an indeterminate
> to a ring; such rings are UFDs if the original ring was, by the
> generalized Gauss Lemma).
> My earlier query about rings with a transfinite Euclidean algorithm is
> separate from this, because such rings are PIDs (by the usual proof,
> using the algorithm to find a "GCD" of two elements in the ideal they
> generate).
> -- JS
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