Abstract:
Zero-sum problems over finite abelian groups have been studied extensively over the last decade for their application to factorization theory. In order to understand the factorization properties of an algebraic number field, one must completely understand the structure of all minimal zero-sum sequences of maximal length. The maximal length for a minimal zero-sum sequence in a finite albelian group, G, is defined by the Davenport constant, which is known only for specific types of groups such as cyclic groups, groups of rank 2, and all p-groups. The minimal zero-sum sequences of maximal length over G = Zsubscript n] have been completely determined. Much progress has been made in the case when G = Z[subscript n] [exclusive or] Z[subscript n], however there is one small gap to be filled. The goal of this paper is to to fill the gap slightly by showing which multiplicities occur in minimal zero-sum sequences over G = Z[subscript n] [exclusive or] Z[subscript n] for n an odd prime. In the search for a proof of the main result, we come across an extension of a well know result in linear Diophantine equations. Given n₁, n₂, ... , n[ subscript k], n [is an element of] Z, it is known that the linear Diophantine equation n₁y₁ + n₂y₂ + ... +n[subscript k]y[subscript k] [is congruent to] 1 mod n has a solution if and only if gcd(n₁, n₂,..., n[subscript k], n) = 1. We look when it is possible for y[subscript i] y[subscript j] mod n for i [is not equal to] j. A proof for n prime is found.
