[FOM] UFD example?

Messing messing at math.umn.edu
Mon Nov 19 12:33:49 EST 2007

Any regular local ring is an unique factorization domain.  The local
ring at a point x, of a non-singular algebraic variety defined over a
field k is such a ring.  If R is an unique factorization domain and S is
a multiplicatively stable subset not containing 0, then S^{-1}R, the
ring of fractions whose denominators are in S, is also an unique
factorization domain.  If R is a field or more generally a principal
ideal domain, the the ring of formal power series R[[X_1, ..., X_n]] is
an unique factorization domain.  If R is the field of real numbers, 
X,Y.Z are variables then R[X, Y, Z]/(X^2 + Y^2 + Z^2 - 1) is an unique 
factorization domain, but the analogous ring where the real field is 
replaced by the complex field is not an unique factorization domain. 
For all this and much more one can consult Bourbaki's Commutative 
Algebra, chapter 7 and Pierre Samuel's Lectures of Unique Factorization 
Domains, published by the Tata Insittute.

William Messing
School of Mathematics
University of Minnesota

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