DIVISOR SEQUENCES OF ATOMS IN KRULL MONOIDS

The divisor sequence of an irreducible element (atom) a of a reduced monoid H is the sequence (s(n))(n is an element of N) where, for each positive integer n, s(n) denotes the number of distinct irreducible divisors of an. We investigate which sequences of positive integers can be realized as divisor sequences of irreducible elements in Krull monoids. In particular, this gives a means for studying nonunique direct-sum decompositions of modules over local Noetherian rings for which the Krull-Remak-Schmidt property fails.