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1. Introduction. Let S be a sequence of elements from the cyclic group Z m . We say S is zsf (zero-sum free) if there does not exist an m-term subsequence of S whose sum is zero. Let g(m, k) (resp. g * (m, k)) denote the least integer such that every sequence S with at least (resp. with exactly) k distinct elements and length g(m, k) (resp. g * (m, k)) must contain an m-term subsequence whose sum is zero. By an affine transformation in Z m we mean a map of the form x → ax + b, with a, b ∈ Z m and gcd(a, m) = 1. Furthermore, let E(m, s) denote the set of all equivalence classes of zsf sequences S of length s, up to order and affine transformation, that are not a proper subsequence of another zsf sequence. Using the above notation, the renowned Erdős-Ginzburg-Ziv Theorem (The function g(m, k) was introduced in [4], where it was shown that g(m, 4) = 2m − 3 for m ≥ 4. Furthermore, based on a lower bound construction the authors conjectured the value of g(m, k) for fixed k and sufficiently large m. Concerning the upper bound, they established an upper bound for m prime modulo the affirmation of the Erdős-Heilbronn conjecture (EHC). Since then, the EHC has been affirmed [9], [2], moreover, the bound given in [4] was extended for nonprimes in [19]. As will later be seen, it is worthwhile to mention that the affirmation of the EHC has resulted in several attempted generalizations and related results [6]

Let G be a finite abelian group, and let n be a positive integer. From the Cauchy-Davenport Theorem it follows that if G is a cyclic group of prime order, then any collection of n subsets A1, A2, . . . , An of G satisfiesM. Kneser generalized the Cauchy-Davenport Theorem for any abelian group. In this paper, we prove a sequence-partition analog of the Cauchy-Davenport Theorem along the lines of Kneser's Theorem. A particular case of our theorem was proved by J. E. Olson in the context of the Erdős-Ginzburg-Ziv Theorem.

Abstract. Let R be a ring and let C be a small class of right R-modules which is closed under finite direct sums, direct summands, and isomorphisms. Let V(C) denote a set of representatives of isomorphism classes in C and, for any module M in C, let [M ] denote the unique element in V(C) isomorphic to M . Then V(C) is a reduced commutative semigroup with operation defined by [M ] , and this semigroup carries all information about direct-sum decompositions of modules in C. 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 M ∈ C, then V(C) is a Krull monoid. Suppose that the monoid V(C) is Krull with a finitely generated class group (for example, when C 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 V(C) 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 V(C) for certain classes of modules over Prüfer rings and hereditary Noetherian prime rings.

Let H be a Krull monoid with finite class group G such that every class contains a prime divisor and let sans-serifD(G) be the Davenport constant of G. Then a product of two atoms of H can be written as a product of at most sans-serifD(G) atoms. We study this extremal case and consider the set scriptU{2,D(G)}(H) defined as the set of all l∈double-struckN 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 sans-serifD(G) atoms. If G is cyclic, then scriptU{2,D(G)}(H)={2,D(G)}. If G has rank two, then we show that (apart from some exceptional cases) scriptU{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.

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 c(H) of H is the smallest integer N with the following property: for each a ∈ H and each two factorizations z, z ′ of a, there exist factorizations z = z 0 , . . . , z k = z ′ of a such that, for each i ∈ [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. Under a very mild condition on the Davenport constant of G, we establish a new and simple characterization of the catenary degree. This characterization gives a new structural understanding of the catenary degree. In particular, it clarifies the relationship between c(H) and the set of distances of H and opens the way towards obtaining more detailed results on the catenary degree. As first applications, we give a new upper bound on c(H) and characterize when c(H) ≤ 4.

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