y in (q2 n ... n q) -p we have that (xj): y = p. Since x; is not in Up and hd p < 1, it follows from the previous proposition that p = (xj). THEOREM 3. Let R be a local domain of dimension <S such that hd R/p < co for all minimal prime ideals p. Then R is a unique factorization domain. Proof: Since R is a noetherian domain, it follows from reference 4; Lemma 1, pg. 408, that it suffices to show that each minimal prime ideal is principal in order to show that R is a unique factorization domain. But by Corollary 2, it will follow that a minimal prime ideal p is principal if we can show that hd Rip < 2. Since hd RIp < a) we have by reference 1; 3.7 and 1.3 that hd R/p + Codim R/ip = Codim R < dim R. But Codim RIp > 1 and dim R < 3. Thus hd R/p < 2, which completes the proof. Since every module has finite homological dimension over a regular local ring, we have established COROLLARY 4. Every regular local ring of dimension <3 is a unique factorization domain. THEOREM 5. Every regular local ring is a unique factorization domain.