Unique Factorization in Regular Local Rings.

In this note we prove that every regular local ring of dimension 3 is a unique factorization domain. Nagata4 showed (Proposition 11) that if every regular local ring of dimension 3 is a unique factorization domain, then every regular local ring has unique factorization.* Thus, combining these results we have that every regular local ring is a unique factorization domain. Throughout this note R is a local ring with maximal ideal 9)1. The definitions and notation follow those of Auslander and Buchsbaum.' PROPOSITION 1. Let x be in 9N and y in R such that a = (x): y satisfies the following conditions: (a) Md a < 1 and (b) x is not in Ma. Then a = (x) and x is not a zero divisor. generating system for a, we have that f is an epimorphism and K is contained in 9fRtn+1. From the exact sequence 0-a K-R n+,-a 0 and the fact that hd a < 1, we have that K is R-free. Since a is not principal, we have that hd a = 1 and thus K $ 0. Let Ni = (t40. .., tin) be a free basis for K over R (i = 1, ...,m). Since a, is in a, we have that yal =-vxfor some v in R. m j=lm j= Now xV =-yT. Therefore we have that E xrjNj = E-ysjNj. Since the j=1 j=1 Nj,} are a free basis for K over R, we have that xrj =-ysj for all j = 1, ..., m. But (x) :y = a. Therefore we have that each sj is in a. Since T = (a,,-x, 0, m m ..., 0) = Ej sjNj, it follows that-x = E sjtjl. Therefore x is in Ma (since K j=1 j=1 is contained in 9YR7+') which contradicts the fact that x is not in Va. Thus a = (x). The fact that x is not a zero divisor follows from the assumption hd a < 1 < o and [2, corollary 6.3]. COROLLARY 2. Suppose p is a prime ideal in R such that dim R, = 1 and hd R/p < 2. Then p is a principal ideal. Proof: Since hdRR/p > hdR5R/pR5 = gi. dim R., it follows that the gi. dim Rp is finite. (See reference 1, 1.6 and reference 3; VIII, 2.6'.) Therefore R. is a regular local ring of dimension 1 (see …