[FOM] UFD example?

Timothy Y. Chow tchow at alum.mit.edu
Mon Nov 19 12:56:56 EST 2007

Joe Shipman wrote:
> What is an example of Unique Factorization Domain which is neither a 
> Principal Ideal Domain nor (isomorphic to) a polynomial ring?

You may want to look at section 20 of Matsumura's book "Commutative Ring 
Theory," which has results such as the following.

Theorem 20.1.  A Noetherian integral domain A is a UFD if and only if 
every height 1 prime ideal is principal.

Theorem 20.3 (Auslander-Buchsbaum).  A regular local ring is a UFD.

Theorem 20.8.  If A is a regular UFD then so is A[[X]] (the formal power 
series ring over A).

In particular, I think that a power series ring in two variables over a 
field is an example of what you're looking for.


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