Extensions of Half-Factorial Domains

Abstract

A half-factorial domain (HFD) is an atomic domain, R, with the property that if one has the irreducible factorizations in RHalf-factorial domains were implicitly studied in the case of rings of algebraic integers in a 1960 paper of L. Carlitz [11] and were subsequently abstracted and studied by A. Zaks.Since their inception as a generalization of the classical notion of unique factorization domains, half-factorial domains have been the subject of much interest in commutative algebra. This paper will give a survey of some recent advances in the study of half-factorial domains with the emphasis on advances in the understanding of ring extension behavior of half-factorial domains.

Arithmetic of non-principal orders in algebraic number fields

“…Half-factoriality has been a central topic ever since the beginning of factorization theory (see the surveys [5,8,22], and [6,7,9,17] for some recent results). A classical result due to Carlitz states that a ring of integers, i.e., a principal order, is half-factorial if and only if its class group has at most two elements (see [2]; there are analogous results for Krull monoids, but for simplicity we restrict our discussion here to rings of integers).…”