Unique factorization of *-products of one-fibered monomial ideals

Abstract

Let R k[x ₁,x ₂,x ₃] be a 3-dimensional polynomial ring over a field k.We show that no non-trivial relations exist among *-products of complete one-fibered (x ₁,x ₂,x ₃)-primary monomial ideals. This is connected with questions about how Zariski's work on the unique factorization of complete ideals in regular local rings of dimension two might be generalized to higher dimensions.