Non-unique factorizations: A survey

Abstract

It is well known that the ring of integers of an algebraic number field may fail to have unique factorization. In the development of algebraic number theory in the 19th century, this failure led to Dedekind's ideal theory and to Kronecker's divisor theory. Only in the late 20th century, starting with L. Carlitz' result concerning class number 2, W. Narkiewicz began a systematic combinatorial and analytic investigation of phenomena of non-unique factorizations in rings of integers of algebraic number fields (see Chapter 9 of [24] for a survey of the early history of the subject). In the sequel several authors started to investigate factorization properties of more general integral domains in the spirit of R. Gilmer's book [18] (see for example the series of papers [2], [3], [4] and the survey article [19] by R. Gilmer). It soon turned out that the investigation of factorization problems can successfully be carried out in the setting of commutative cancellative monoids, and this point of view opened the door to further applications of the theory. Among them the most prominent ones are the arithmetic of congruence monoids, the theory of zero-sum sequences over abelian groups and the investigation of Krull monoids describing the deviation from the Krull-Remak-Azumaya-Schmidt Theorem in certain categories of modules. The proceedings [1] and [6] of two Conferences on Factorization Theory (held 1996 in Iowa City and 2003 in Chapel Hill) and the articles contained in [7] give a good survey on the development of the theory of non-unique factorizations over the past decade. Only recently the authors completed the monograph [16] which contains a thorough presentation of the algebraic, combinatorial and analytic aspects of the theory of non-unique factorizations, together with self-contained introductions to additive group theory, to the theory of u-ideals and to abstract analytic number theory. The purpose of this survey article is to point out some highlights of the theory of non-unique factorizations (Theorem 5.6.B. and Theorem 7.4) with an emphasis on the presentation of the concepts which describe the various phenomena of non-uniqueness in structures of arithmetical relevance. We concentrate on the presentation of the main results and, if at all, we only give the main ideas of the proofs. For more details we refer the reader to the monograph [16] and to the original papers in the volumes cited above.