Factorization Sets and Half-Factorial Sets in Integral Domains Original Research Article

Let R be an atomic integral domain. Suppose that image is a nonempty subset of irreducible elements of R, u is a unit of R, and α1,..., αn, β1,..., βm are irreducible elements of R such that (1) α1… αn = u · β1… βm. Set imageα = {iαi set membership, variant image} and imageβ = {jβj set membership, variant image}. image is a factorization set (F-set) of R if for any equality involving irreducibles of the form (1), imageα ≠ 0 implies that imageβ ≠ 0. image is a half-factorial set (HF-set) if any equality of the form (1) implies that imageα = imageβ. 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 image(R) the set of irreducible elements of R, then set imageP = P ∩ image(R). We show that if image is an F-set of R, then there exists a nonempty set X of nonzero prime ideals of R such that image = union or logical sum P set membership, variant XimageP. 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 image is an HF-set of R and P is a nonprincipal prime ideal of R with imageP subset of or equal to image, then imageQ subset of or equal to image 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 image(R) is the set of prime elements of R, then image an HF-set of R implies that either image union or logical sum image(R) = image(R) or image subset of or equal to image(R).