Some Questions on Unique

The simplest type of integral extension of a unique factorization domain B is obtained by adjoining to B an n-th root of one of its elements; that is, by forming the ring Bn = B[z]=(z n g), where n 2 Z and g 2 B. When n = 1, Bn = B. When n 2 it is natural to ask for what g 2 B will unique factorization be preserved? The case where B is a polynomial ring over an algebraically closed eld k immediately comes to mind. In the one variable case we have a de nitive answer: Bn is factorial if and only if g 2 B = k[x] is linear; for if deg(g) 2 then g factors completely in B as a product of linear factors and in Bn g will have two distinct factorizations. With more variables the issue becomes more cloudy. This paper concentrates on the two variable case so that our principal question is