Factorization Sets and Half-Factorial Sets in Integral Domains

Abstract

Let R be an atomic integral domain. Suppose that H is a nonempty subset of irreducible elements of R, u is a unit of R, and α 1,..., α n , β 1,..., β m are irreducible elements of R such that (1) α 1... α n = u · β 1... β m . Set H α = { i|α i ∈ H } and H β = { j|β j ∈ H }. H is a factorization set ( F-set) of R if for any equality involving irreducibles of the form (1), | H α| ≠ 0 implies that | H β| ≠ 0. H is a half-factorial set ( HF-set) if any equality of the form (1) implies that | H α| = | H β|. In this paper, we explore in detail the structure of the F-sets and HF-sets of an atomic integral domain R. If P is a nonzero prime ideal of R and J ( R) the set of irreducible elements of R, then set H P = P ∩ J ( R). We show that if F is an F-set of R, then there exists a nonempty set X of nonzero prime ideals of R such that F = ∪ P ∈ X H P . We define an F-set to be minimal if it contains no proper subsets which are F-sets. We then show that in R every F-set can be written as a union of minimal F-sets if and only if R satisfies the Principal Ideal Theorem. If, in addition, every height-one prime ideal of R is the radical of a principal ideal, then this representation is unique. We study the structure of the HF-sets of R and concentrate on the case where R is a Krull domain with torsion divisor class group. For such a domain R, we show that if H is an HF-set of R and P is a nonprincipal prime ideal of R with H P ⊆ H , then H Q ⊆ H for each prime ideal Q of R in the same divisor class as P. We also show that if R is a Dedekind domain with prime class number p ≥ 2 and P ( R) is the set of prime elements of R, then H an HF-set of R implies that either H ∪ P ( R) = J ( R) or H ⊆ P ( R).