[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.
Tim
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