Paul Baginski

Products of two atoms in Krull monoids and arithmetical characterizations of class groups

Let H be a Krull monoid with finite class group G such that every class contains a prime divisor and let D(G) be the Davenport constant of G. Then a product of two atoms of H can be written as a product of at most D(G) atoms. We study this extremal case and consider the set U{2,D(G)}(H) defined as the set of all l∈N with the following property: there are two atoms u,v∈H such that uv can be written as a product of l atoms as well as a product of D(G) atoms. If G is cyclic, then U{2,D(G)}(H)={2,D(G)}. If G has rank two, then we show that (apart from some exceptional cases) U{2,D(G)}(H)=[2,D(G)]∖{3}. This result is based on the recent characterization of all minimal zero-sum sequences of maximal length over groups of rank two. As a consequence, we are able to show that the arithmetical factorization properties encoded in the sets of lengths of a rank 2 prime power order group uniquely characterizes the group.

Arithmetic Congruence Monoids: A Survey

We consider multiplicative monoids of the positive integers defined by a single congruence. If a and b are positive integers such that a ≤ b and \(a^{2} \equiv a\mod b\), then such a monoid (known as an arithmetic congruence monoid or an ACM) can be described as \(M_{a,b} = (a + b\mathbb{N}_{0}) \cup \{ 1\}\). In lectures on elementary number theory, Hilbert demonstrated to students the utility of the proof of the Fundamental Theorem of Arithmetic for \(\mathbb{Z}\) by considering the arithmetic congruence monoid with a = 1 and b = 4. In M 1, 4, the element 441 has a nonunique factorization into irreducible elements as 9 ⋅ 49 = 212. ACMs have appeared frequently in the mathematical literature over the last decade. While their structures can be understood merely with rational number theory, their multiplicative behavior can become quite complex. We show that all ACMs fall into one of three mutually exclusive classes: regular (relating to a = 1), local (relating to \(\gcd (a,b) = p^{k}\) for some rational prime p), and global (\(\gcd (a,b)\) is not a power of a prime). In each case, we examine the behavior of various invariants widely studied in the theory of nonunique factorizations. Our principal tool will be the construction of transfer homomorphisms from the M a, b to monoids with simpler multiplicative structure.

Finding Elements With Given Factorization Lengths and Multiplicities

Abstract Many algebraic number rings exhibit nonunique factorization of elements into irreducibles. Not only can the irreducibles in the factorizations be different, but the number of irreducibles in the factorizations can also vary. A basic question then is: Which sets can occur as the set of factorization lengths of an element? Moreover, how often can each factorization length occur? While these questions are most pertinent in algebraic number rings, their pertinence extends to Dedekind domains and a broader class of structures called Krull monoids. Surprisingly, for a large subclass of Krull monoids, Kainrath was able to resolve completely the question of which length sets and length multiplicities can be realized. In this article, we explain the context of Kainraths theorem and give a constructive proof for an important case, namely Krull monoids with infinite nontorsion class group. We also construct length sets in a case not covered by Kainraths theorem to illustrate the difficulty of the general problem.