FACTORIZATION SETS AND HALF-FACTORIAL SETS IN INTEGRAL-DOMAINS

Let R be an atomic integral domain. Suppose that H is a nonempty subse t of irreducible elements of R, u is a unit of R, and alpha(1),...,alp ha(n), beta(1),...,beta(m) are irreducible elements of R such that (1) alpha(1)...alpha(n) = u .beta(1)...beta(m). Set H-alpha = {i\alpha(i) is an element of H} and H-beta = {j\beta(j) is an element of H}. H is a factorization set (F-set) of R if for any equality involving irredu cibles of the form (1), \H-alpha\ not equal 0 implies that \H-beta\ no t equal 0. H is a half-factorial set (HF-set) if any equality of the f orm (1) implies that \H-alpha\ = \H-beta\. In this paper, we explore i n detail the structure of the F-sets and I-IF-sets of an atomic integr al domain R. If P is a nonzero prime ideal of R and I(R) the set of ir reducible elements of R, then set H-P = P boolean AND I(R). We show th at if F is an F-set of R, then there exists a nonempty set X of nonzer o prime ideals of R such that F = UP is an element of XHP. We define a n F-set to be minimal if it contains no proper subsets which are F-set s. We then show that in R every F-set can be written as a union of min imal 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 p rincipal ideal, then this representation is unique. We study the struc ture of the HF-sets of R and concentrate on the case where R is a Krul l domain with torsion divisor class group. For such a domain R, we sho w that if H is an KF-set of R and P is a nonprincipal prime ideal of R with H-P subset of or equal to H, then H-Q subset of or equal to H fo r 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 greater than or equal to 2 and P(R) is the set of prime elements of R, then H an H F-set of R implies that either H boolean OR P(R) = I(R) or H subset of or equal to P(R). (C) 1995 Academic Press, Inc.