On transfer homomorphisms of Krull monoids

Every Krull monoid has a transfer homomorphism onto a monoid of zero-sum sequences over a subset of its class group. This transfer homomorphism is a crucial tool for studying the arithmetic of Krull monoids. In the present paper, we strengthen and refine this tool for Krull monoids with finitely generated class group.

Arithmetic of commutative semigroups with a focus on semigroups of ideals and modules

Let [Formula: see text] be a commutative semigroup with unit element such that every non-unit can be written as a finite product of irreducible elements (atoms). For every [Formula: see text], let [Formula: see text] denote the set of all [Formula: see text] with the property that there are atoms [Formula: see text] such that [Formula: see text] (thus, [Formula: see text] is the union of all sets of lengths containing [Formula: see text]). The Structure Theorem for Unions states that, for all sufficiently large [Formula: see text], the sets [Formula: see text] are almost arithmetical progressions with the same difference and global bound. We present a new approach to this result in the framework of arithmetic combinatorics, by deriving, for suitably defined families of subsets of the non-negative integers, a characterization of when the Structure Theorem holds. This abstract approach allows us to verify, for the first time, the Structure Theorem for a variety of possibly non-cancellative semigroups, including semigroups of (not necessarily invertible) ideals and semigroups of modules. Furthermore, we provide the very first example of a semigroup (actually, a locally tame Krull monoid) that does not satisfy the Structure Theorem.

THE CATENARY DEGREE OF KRULL MONOIDS II

Let$H$be a Krull monoid with finite class group$G$such that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field). The catenary degree$\mathsf{c}(H)$of$H$is the smallest integer$N$with the following property: for each$a\in H$and each pair of factorizations$z,z^{\prime }$of$a$, there exist factorizations$z=z_{0},\dots ,z_{k}=z^{\prime }$of$a$such that, for each$i\in [1,k]$,$z_{i}$arises from$z_{i-1}$by replacing at most$N$atoms from$z_{i-1}$by at most$N$new atoms. To exclude trivial cases, suppose that$|G|\geq 3$. Then the catenary degree depends only on the class group$G$and we have$\mathsf{c}(H)\in [3,\mathsf{D}(G)]$, where$\mathsf{D}(G)$denotes the Davenport constant of$G$. The cases when$\mathsf{c}(H)\in \{3,4,\mathsf{D}(G)\}$have been previously characterized (see Theorem A). Based on a characterization of the catenary degree determined in the paper by Geroldingeret al.[‘The catenary degree of Krull monoids I’,J. Théor. Nombres Bordeaux23(2011), 137–169], we determine the class groups satisfying$\mathsf{c}(H)=\mathsf{D}(G)-1$. Apart from the extremal cases mentioned, the precise value of$\mathsf{c}(H)$is known for no further class groups.

A semigroup-theoretical view of direct-sum decompositions and associated combinatorial problems

Let R be a ring and let [Formula: see text] be a small class of right R-modules which is closed under finite direct sums, direct summands, and isomorphisms. Let [Formula: see text] denote a set of representatives of isomorphism classes in [Formula: see text] and, for any module M in [Formula: see text], let [M] denote the unique element in [Formula: see text] isomorphic to M. Then [Formula: see text] is a reduced commutative semigroup with operation defined by [M] + [N] = [M ⊕ N], and this semigroup carries all information about direct-sum decompositions of modules in [Formula: see text]. This semigroup-theoretical point of view has been prevalent in the theory of direct-sum decompositions since it was shown that if End R(M) is semilocal for all [Formula: see text], then [Formula: see text] is a Krull monoid. Suppose that the monoid [Formula: see text] is Krull with a finitely generated class group (for example, when [Formula: see text] is the class of finitely generated torsion-free modules and R is a one-dimensional reduced Noetherian local ring). In this case, we study the arithmetic of [Formula: see text] using new methods from zero-sum theory. Furthermore, based on module-theoretic work of Lam, Levy, Robson, and others we study the algebraic and arithmetic structure of the monoid [Formula: see text] for certain classes of modules over Prüfer rings and hereditary Noetherian prime rings.

MAXIMAL IDEALS IN PREADDITIVE CATEGORIES AND SEMILOCAL CATEGORIES

The first aim of this article is to study maximal ideals of a preadditive category [Formula: see text]. Maximal ideals, which do not exist in general for arbitrary preadditive categories, are associated to a maximal ideal of the endomorphism ring of an object and always exist when the category is semilocal. If [Formula: see text] is additive and semilocal, any skeleton [Formula: see text] of [Formula: see text] is a Krull monoid and we are able to characterize the essential valuations of [Formula: see text] and provide some natural divisor homomorphisms and divisor theories of [Formula: see text].